High-dimensional finite elements for multiscale Maxwell-type equations

High-dimensional finite elements for multiscale Maxwell-type equations Abstract We consider multiscale Maxwell-type equations in a domain $$D\subset\mathbb{R}^d$$ ($$d=2,3$$), which depend on $$n$$ microscopic scales. Using multiscale convergence, we derive the multiscale homogenized problem, which is posed in $$\mathbb{R}^{(n+1)d}$$. Solving it, we get all the necessary macroscopic and microscopic information. Sparse tensor product finite elements (FEs) are employed, using edge FEs. The method achieves a required level of accuracy with essentially an optimal number of degrees of freedom, which, apart from a multiplying logarithmic term, is equal to that for solving a problem in $$\mathbb{R}^d$$. Numerical correctors are constructed from the FE solutions. In the two-scale case, an explicit homogenization error is deduced. To get this error, the standard procedure in the homogenization literature requires the solution $$u_0$$ of the homogenized problem to belong to $$H^1({\rm curl\,},D)$$. However, in polygonal domains, $$u_0$$ belongs only to a weaker regularity space $$H^s({\rm curl\,},D)$$ for $$0<s<1$$. We derive a homogenization error estimate for this case. Though we prove the result for two-scale Maxwell-type equations, the approach works verbatim for elliptic and elasticity problems when the solution to the homogenized equation belongs to $$H^{1+s}(D)$$ (standard procedure requires $$H^2(D)$$ regularity). This homogenization error estimate is new in the literature. Thus, for two-scale problems, an explicit error for the numerical corrector is obtained; it is of the order of the sum of the homogenization error and the FE error. For the case of more than two scales, we construct a numerical corrector, albeit without a rate of convergence, as such a homogenization error is not available. Numerical experiments confirm the theoretical results. 1. Introduction We consider Maxwell-type equations that depend on $$n$$ separable microscopic scales in a domain $$D\in \mathbb{R}^d,$$ where $$d=2,3$$. The coefficients are assumed to be locally periodic with respect to each microscopic scale. We use the multiscale convergence of a bounded sequence in $$H({\rm curl\,},D)$$ to derive the multiscale homogenized equation, which contains all the necessary information. Solving it, we get the solution of the homogenized equation that describes the multiscale solution macroscopically and the scale interacting terms (the corrector terms) that encode the multiscale information. However, this equation is posed in high-dimensional product domains. It depends on $$n+1$$ variables in $$\mathbb{R}^d$$, one for each scale that the original multiscale problem depends on. The full tensor product finite element (FE) method requires a large number of degrees of freedom, and thus is prohibitively expensive. We develop the sparse tensor FE product approach, using edge FEs, for this multiscale homogenized Maxwell-type equation. The approach achieves accuracy essentially equal to that obtained by the full tensor product FEs but requires an essentially optimal level of complexity that is essentially equal to that for solving a problem in $$\mathbb{R}^d$$ only. Analytic homogenization for two-scale Maxwell-type equations is well developed. We mention the standard references Bensoussan et al. (1978), Sanchez-Palencia (1980) and Jikov et al. (1994). However, there has been little effort on numerical analysis of multiscale Maxwell-type equations. As for other multiscale problems, a direct numerical treatment needs a fine mesh which is at most of the order of the smallest scale, leading to a prohibitive level of complexity. The multiscale FE method (Hou & Wu, 1997; Efendiev & Hou, 2009) and the heterogeneous multiscale method (E & Engquist, 2003; Abdulle et al., 2012) are designed to overcome this difficulty but their applications to multiscale Maxwell-type equations have not been adequately studied. Solving cell problems to establish the homogenized equation and using the cell problems’ solutions to compute the correctors for two-scale Maxwell-type equations are performed in Zhang et al. (2010). However, as for other multiscale problems, this approach is rather expensive, especially when the coefficients are only locally periodic, as for each macroscopic point, several cell problems need to be solved. We contribute in this article a feasible general numerical method for locally periodic multiscale Maxwell-type problems. We employ the sparse tensor product FE approach developed by Hoang & Schwab (2004/05) for multiscale elliptic equations (see also Hoang, 2008; Harbrecht & Schwab, 2011; Xia & Hoang, 2014, 2015a,b). It achieves the required level of accuracy with an essentially optimal number of degrees of freedom. We note that sparse tensor edge FEs are considered in Hiptmair et al. (2013) in the context of computing the moments of the solutions to stochastic Maxwell-type problems. However, our setting is quite different, and does not require constructing the detail spaces for edge FEs. We only need the detail spaces for the nodal FEs that approximate functions in the Lebesgue spaces $$L^2$$. We then construct a numerical corrector for the solution of the original multiscale problem, using the FE solutions of the multiscale homogenized problem. In the case of two scales, we derive an explicit error estimate in terms of the homogenization error and the FE error. It is well known that for two-scale elliptic problems in a domain $$D$$, if the solution of the homogenized problem belongs to $$H^2(D)$$, and the solutions to the cell problems are sufficiently smooth, the homogenization error in the $$H^1(D)$$ norm is $${\mathcal O}(\varepsilon^{1/2}),$$ where $$\varepsilon$$ is the microscopic scale (Bensoussan et al., 1978; Jikov et al., 1994). For two-scale Maxwell-type equations, the $${\mathcal O}(\varepsilon^{1/2})$$ homogenization error in the $$H({\rm curl},D)$$ norm is obtained when the solution $$u_0$$ of the homogenized problem (4.7) belongs to $$H^1({\rm curl},D)$$. However, for polygonal domains that are of interest in FE discretization, $$u_0$$ generally belongs only to a weaker regularity space $$H^s({\rm curl\,},D)$$ for $$0<s<1$$ (see e.g., Hiptmair, 2002). For this case, we develop an approach to deriving a new homogenization error estimate. Though we present the result for Maxwell-type equations, the approach works verbatim for two-scale elliptic and elasticity problems when the solution to the homogenized problem is in $$H^{1+s}(D)$$. As far as we are aware, this is a new result in the homogenization theory and forms another main contribution of the article. For the case of more than two scales, an analytic homogenization error is not available. However, we can still derive a corrector from the FE solution of the multiscale homogenized problem, albeit without an explicit rate of convergence. This article is organized as follows. In the next section, we formulate the multiscale Maxwell-type equation. Homogenization of the multiscale Maxwell-type equation (2.4) is studied in Bensoussan et al. (1978) in the two-scale case, using two-scale asymptotic expansion. Here, we use the multiscale convergence method to study (2.4) in the general multiscale setting. We thus develop multiscale convergence for a bounded sequence in $$H({\rm curl\,},D)$$. Two-scale convergence for a bounded sequence in $$H({\rm curl\,},D)$$ is developed in Wellander & Kristensson (2003). Since we consider the general $$(n+1)$$-scale convergence and the limiting result that we use is in a slightly different form from that of Wellander & Kristensson (2003) in the two-scale case, so we present the proofs in full. FE approximations of the multiscale homogenized Maxwell-type problem are studied in Section 3. We prove the FE error estimates in cases of both full and sparse tensor product FE approximations. The errors are essentially equal (apart from a logarithmic multiplying factor), but the dimension of the sparse tensor product FE space is much lower than that of the full tensor product FE space and is essentially equal to that for solving a problem in $$\mathbb{R}^d$$ only. In Section 4, we construct numerical correctors for the solution to the original multiscale problem. For two-scale problems, we prove the general homogenization error estimate for the case where $$u_0$$ belongs to the weaker regularity space $$H^s({\rm curl\,},D),$$ where $$0<s<1$$. From that we deduce the error estimate for the numerical corrector, which is of the order of the sum of the homogenization error estimate and the FE error. For the case of more than two scales, we derive a numerical corrector but without a rate of convergence. In Section 5, we prove that the regularity required to get the FE error estimate for the sparse tensor product FEs and to get the homogenization error in the two-scale case is achievable. Section 6 contains numerical experiments that confirm our analysis. Finally, the two Appendices A and B contain the long proofs of some previous results: the proof of the homogenization error when $$u_0$$ belongs to a weaker regularity space is presented in Appendix A. Throughout the article, by $$\#$$ we denote the spaces of functions that are periodic with the period being the unit cube $$Y\subset \mathbb{R}^d$$. Repeated indices indicate summation. The notations $$\nabla$$ and $${\rm curl\,}$$ without indicating the variable explicitly denote the gradient and the $${\rm curl\,}$$ operator with respect to $$x$$ of a function of $$x$$ only, where $$\nabla_x$$ and $${\rm curl\,}_{\!x}$$ denote the partial gradient and partial $${\rm curl\,}$$ of a function depending on $$x$$ and also on other variables. We generally present the theoretical results for the three-dimensional case and mention the two-dimensional case only when it is necessary, as the two cases are largely similar. 2. Problem setting 2.1 Multiscale Maxwell-type problems Let $$D$$ be a domain in $$\mathbb{R}^d$$ ($$d=2,3$$). Let $$Y$$ be the unit cube in $$\mathbb{R}^d$$. By $$Y_1,\ldots,Y_n$$ we denote $$n$$ copies1 of $$Y$$. We denote by $${\bf Y}$$ the product set $$Y_1\times Y_2\times\cdots\times Y_n$$ and by $$\boldsymbol{y}\in{\bf Y}$$ the vector $$\boldsymbol{y}=(y_1,y_2,\ldots,y_n)$$. For each $$i=1,\ldots,n$$, we denote by $${\bf Y}_i$$ the set of vectors $$\boldsymbol{y}_i=(y_1,\ldots,y_i),$$ where $$y_j\in Y_j$$ for $$j=1,\ldots,i$$. For $$d=3$$, let $$a$$ and $$b$$ be functions with symmetric matrix values from $$D\times {\bf Y}$$ to $$\mathbb{R}^{d\times d}_{\rm sym}$$; $$a$$ and $$b$$ are continuous in $$D\times {\bf Y}$$ and are periodic with respect to each variable $$y_i$$ with the period being $$Y_i$$. We assume that for all $$x\in D$$ and $$\boldsymbol{y}\in{\bf Y}$$, and all $$\xi,\zeta\in \mathbb{R}^d$$,   c∗|ξ|2≤aij(x,y)ξiξj,   aij(x,y)ξiζj≤c∗|ξ||ζ|,c∗|ξ|2≤bij(x,y)ξiξj,   bij(x,y)ξiζj≤c∗|ξ||ζ|, (2.1) where $$c_*$$ and $$c^*$$ are positive numbers; $$|\cdot|$$ denotes the Euclidean norm in $$\mathbb{R}^3$$. Let $$\varepsilon$$ be a small positive value, and $$\varepsilon_1,\ldots,\varepsilon_n$$ be $$n$$ functions of $$\varepsilon$$ that denote the $$n$$ microscopic scales that the problem depends on. We assume the following scale separation properties: for all $$i=1,\ldots,n-1$$,   limε→0εi+1(ε)εi(ε)=0. (2.2) Without loss of generality, we assume that $$\varepsilon_1=\varepsilon$$. We define $$a^\varepsilon, b^\varepsilon: D\to\mathbb{R}^{d\times d}_{\rm sym}$$ as   aε(x)=a(x,xε1,…,xεn),  bε(x)=b(x,xε1,…,xεn). (2.3) Let   W=H0(curl,D)={u∈L2(D)3,  curlu∈L2(D)3,  u×ν=0}, where $$\nu$$ denotes the outward normal vector on the boundary $$\partial D$$. Let $$f\in W'$$. We consider the problem   curl(aε(x)curluε(x))+bε(x)uε(x)=f(x), (2.4) with the boundary condition $$u^{\varepsilon}\times \nu=0$$ on $$\partial D$$. We formulate this problem in the variational form as follows: find $$u^{\varepsilon}\in W$$ so that   ∫D[aε(x)curluε(x)⋅curlϕ(x)+bε(x)uε(x)⋅ϕ(x)]dx=∫Df(x)⋅ϕ(x)dx (2.5) for all $$\phi\in W$$ (by $$\int_D\,f\cdot\phi\, {\rm d}x$$ we denote the duality pairing between $$W'$$ and $$W$$). The Lax–Milgram lemma guarantees the existence of a unique solution $$u^{\varepsilon}$$ that satisfies   ‖uε‖W≤c‖f‖W′, (2.6) where the constant $$c$$ depends only on $$c_*$$ and $$c^*$$ in (2.1). For $$d=2$$, the matrix function $$b^\varepsilon:D\times{\bf Y}\to \mathbb{R}^{2\times 2}$$ is defined as above. As $${\rm curl\,}u^{\varepsilon}$$ is now a scalar function, $$a(x,\boldsymbol{y})$$ is a continuous function from $$D\times{\bf Y}$$ to $$\mathbb{R},$$ which is periodic with respect to each variable $$y_i$$ with the period being $$Y_i$$. In the place of (2.1), we have   c∗≤a(x,y)≤c∗  ∀x∈D and y∈Y. The variational formulation in two dimensions becomes   ∫D[aε(x)curluε(x)curlϕ(x)+bε(x)uε(x)⋅ϕ(x)]dx=∫Df(x)⋅ϕ(x)dx  ∀ϕ∈W. (2.7) In the rest of the article, we present the results for the three-dimensional case and only mention the two-dimensional case when necessary; the results for two dimensions are similar. 2.2 Multiscale convergence We use multiscale convergence to derive the homogenized equation. We first recall the definition of multiscale convergence (see Nguetseng, 1989; Allaire, 1992; Allaire & Briane, 1996). Definition 2.1 A sequence of functions $$\{w^\varepsilon\}_\varepsilon\subset L^2(D)$$$$(n+1)$$-scale converges to a function $$w^0\in L^2(D\times {\bf Y})$$ if for all smooth functions $$\phi\in C^\infty(D\times{\bf Y}),$$ which are periodic with respect to $$y_i$$ with the period being $$Y_i$$ for $$i=1,\ldots,n$$,   limε→0∫Dwε(x)ϕ(x,xε1,…,xεn)dx=∫D∫Yw0(x,y)ϕ(x,y)dydx. We have the following result. Proposition 2.2 From a bounded sequence in $$L^2(D),$$ we can extract an $$(n+1)$$-scale convergent subsequence. For a bounded sequence in $$H({\rm curl\,},D)$$, we have the following results on $$(n+1)$$-scale convergence. These results were first established in Wellander & Kristensson (2003) for the two-scale case. We present below the multiscale convergence of a bounded sequence in $$H({\rm curl\,},D),$$ which will be used to study the multiscale equations (2.5) and (2.7). By $$\tilde H_\#({\rm curl\,},Y_i)$$ we denote the equivalent classes of functions in $$H_\#({\rm curl\,},Y_i)$$ such that if $${\rm curl\,} v={\rm curl\,} w$$ we regard $$v=w$$ in $$\tilde H_\#({\rm curl\,},Y_i)$$. Proposition 2.3 Let $$\{w^\varepsilon\}_\varepsilon$$ be a bounded sequence in $$H({\rm curl\,},D)$$. There is a subsequence (not renumbered), a function $$w_0\in H({\rm curl\,},D)$$, $$n$$ functions $${\frak w_i}\in L^2(D\times Y_1\times\cdots\times Y_{i-1},H^1_\#(Y_i)/\mathbb{R})$$ such that   wε⟶(n+1)−scalew0+∑i=1n∇yiwi. Further, there are $$n$$ functions $$w_i\in L^2(D\times Y_1\times\cdots\times Y_{i-1},\tilde H_\#({\rm curl\,},Y_i))$$ such that   curlwε⟶(n+1)−scalecurlw0+∑i=1ncurlyiwi. Proof. Let $$\xi\in L^2(D\times{\bf Y})^3$$ be the $$(n+1)$$-scale limit of $$\{w^\varepsilon\}_\varepsilon$$. Consider the function $$\phi=\varepsilon_n{\it{\Phi}}(x,y_1,\ldots,y_n),$$ where $${\it{\Phi}}$$ is a function in $$C^\infty_0(D,C^\infty_\#(Y_1,\ldots,C^\infty_\#(Y_n),\ldots))^3$$ and is periodic with respect to $$y_1, \ldots,y_n$$ with the period being $$Y_1,\ldots,Y_n,$$ respectively. We then have   limε→0∫Dcurlwε⋅εnΦ(x,xε1,…,xεn)dx=0. On the other hand,   limε→0∫Dcurlwε⋅εnΦ(x,xε1,…,xεn)dx = limε→0∫Dwε⋅εncurlΦ(x,xε1,…,xεn)dx = limε→0∫Dwε⋅curlynΦ(x,xε1,…,xεn)dx = ∫D∫Yξ(x,y)⋅curlynΦ(x,y)dydx. Thus, there is a function $$\xi_{n-1}(x,\boldsymbol{y}_{n-1})\in L^2(D\times{\bf Y}_{n-1})$$ and a function $${\frak w_n}(x,\boldsymbol{y}_n)\in L^2(D\times{\bf Y}_{n-1},H^1_\#(Y_n)/\mathbb{R})$$ such that   ξ(x,y)=ξn−1(x,yn−1)+∇ynwn(x,y). Next we choose $$\phi=\varepsilon_{n-1}{\it{\Phi}}(x,y_1,\ldots,y_{n-1})$$ for a function $${\it{\Phi}}\in C^\infty_0(D,C^\infty_\#(Y_1,\ldots,C^\infty_\#(Y_{n-1}),\ldots),$$ which is periodic with respect to $$y_1,\ldots,y_{n-1}$$. We then have   0 = limε→0∫Dcurlwε⋅εn−1Φ(x,xε1,…,xεn−1)=limε→0∫Dwε⋅curlyn−1Φ(x,xε1,…,xεn−1)dx = ∫D∫Y(ξn−1(x,yn−1)+∇ynwn(x,y))⋅curlyn−1Φ(x,y1,…,yn−1)dyn−1dx = ∫D∫Yn−1ξn−1(x,yn−1)⋅curlyn−1Φ(x,y1,…,yn−1)dyn−1dx. From this, there is a function $$\xi_{n-2}(x,\boldsymbol{y}_{n-2})\in L^2(D\times {\bf Y}_{n-2})$$ and a function $${\frak w_{n-1}}(x,\boldsymbol{y}_{n-1})\in L^2(D\times {\bf Y}_{n-2},H^1_\#(Y_{n-1})/\mathbb{R})$$ so that   ξn−1(x,yn−1)=ξn−2(x,yn−2)+∇yn−1wn−1(x,yn−1), so   ξ(x,y)=ξn−2(x,yn−2)+∇yn−1wn−1(x,yn−1)+∇ynwn(x,y). Continuing this process, we have   ξ(x,y)=w0(x)+∑i=1n∇yiwi(x,yi), where $$w_0\in L^2(D)^3$$ and $${\frak w_i}(x,\boldsymbol{y}_i)\in L^2(D\times {\bf Y}_{i-1},H^1_\#(Y_i))$$. As $$\int_Y\xi(x,\boldsymbol{y})\,{\rm d}\boldsymbol{y}=w_0(x)$$, $$w_0$$ is the weak limit of $$w^\varepsilon$$ in $$L^2(D)^3$$. Let $$\eta(x,\boldsymbol{y})$$ be the $$(n+1)$$-scale convergence limit of $${\rm curl\,} w^\varepsilon$$ in $$L^2(D\times{\bf Y})$$. Let $${\it{\Phi}}(x,y_1,\ldots,y_n)\in C^\infty_0(D,C^\infty_\#(Y_1,\ldots,C^\infty_\#(Y_n),\ldots))$$. We have    ∫Dcurlwε⋅∇Φ(x,xε1,…,xεn)dx  =∫Dwε⋅curl∇Φ(x,xε1,…,xεn)dx−∫∂D(wε×ν)⋅∇Φ(x,xε1,…,xεn)ds=0. Thus,   0 = limε→ 0∫Dcurlwε⋅εn∇Φ(x,xε1,…,xεn)dx=limε→0∫Dcurlwε⋅∇ynΦ(x,xε1,…,xεn)dx = ∫D∫Yη(x,y)⋅∇ynΦ(x,y1,…,yn)dydx. Therefore, there is a function $$w_n(x,\boldsymbol{y}_n)\in L^2(D\times{\bf Y}_{n-1},\tilde H_\#({\rm curl\,},Y_n))$$ and a function $$\eta_{n-1}(x,\boldsymbol{y}_{n-1})\in L^2(D\times{\bf Y}_{n-1})$$ such that   η(x,y)=ηn−1(x,yn−1)+curlynwn(x,y). Let $${\it{\Phi}}(x,y_1,\ldots,y_{n-1})\in C^\infty_0(D,C^\infty_\#(Y_1,\ldots,C^\infty_\#(Y_{n-1}),\ldots))$$. We have   0 = limε→ 0∫Dcurlwε⋅εn−1∇Φ(x,xε1,…,xεn−1)dx=limε→0∫Dcurlwε⋅∇yn−1Φ(x,xε1,…,xεn−1)dx = ∫D∫Y(ηn−1(x,yn−1)+curlynwn(x,y))⋅∇yn−1Φ(x,y1,…,yn−1)dydx = ∫D∫Yn−1ηn−1(x,yn−1)⋅∇yn−1ϕ(x,y1,…,yn−1)dyn−1dx. Therefore, there is a function $$w_{n-1}(x,\boldsymbol{y}_{n-1})\in L^2(D\times{\bf Y}_{n-2},\tilde H_\#({\rm curl\,},Y_{n-1}))$$ and a function $$\eta_{n-2}(x,\boldsymbol{y}_{n-2})\in L^2(D\times {\bf Y}_{n-2})^3$$ so that   ηn−1(x,yn−1)=ηn−2(x,yn−2)+curlyn−1wn−1(x,yn−1) so   η(x,y)=ηn−2(x,yn−2)+curlyn−1wn−1(x,yn−1)+curlynwn(x,y). Continuing, we find that there is a function $$\eta_0(x)\in L^2(D)^3$$ and functions $$w_i(x,\boldsymbol{y}_{i})\in L^2(D\times{\bf Y}_{i-1},\tilde H_\#({\rm curl\,},Y_i))$$ so that   η(x,y)=η0(x)+∑i=1ncurlyiwi(x,yi). As for all $$\phi(x)\in C^\infty_0(D)^3$$  limε→0∫Dcurlwε(x)⋅ϕ(x)dx=∫Dη0(x)⋅ϕ(x)dx,$$\eta_0$$ is the weak limit of $${\rm curl\,} w^\varepsilon$$ in $$L^2(D)^3$$. Thus, $$\eta_0={\rm curl\,} w_0$$. We then get the conclusion. □ 2.3 Multiscale homogenized Maxwell-type problem From (2.6) and Proposition 2.3, we can extract a subsequence (not renumbered), a function $$u_0\in H_0({\rm curl\,}, D)$$, $$n$$ functions $${\frak u_i}\in L^2(D\times Y_1\times\cdots\times Y_{i-1},H^1_\#(Y_i)/\mathbb{R})$$ and $$n$$ functions $$u_i\in L^2(D\times Y_1\times\cdots\times Y_{i-1},\tilde H_\#({\rm curl\,},Y_i))$$ such that   uε⟶(n+1)−scaleu0+∑i=1n∇yiui (2.8) and   curluε⟶(n+1)−scalecurlu0+∑i=1ncurlyiui. (2.9) For $$i=1,\ldots,n$$, let $$W_i=L^2(D\times Y_1\times\cdots\times Y_{i-1},\tilde H_\#({\rm curl\,},Y_i))$$ and $$V_i=L^2(D\times Y_1\times\cdots\times Y_{i-1},H^1_\#(Y_i)/\mathbb{R})$$. We define the space $${\bf V}$$ as   V=W×W1×⋯×Wn×V1×⋯×Vn. For $$\boldsymbol{v}=(v_0,\{v_i\},\{\frak v_i\})\in {\bf V}$$, we define the norm   |||v|||=‖v0‖H(curl,D)+∑i=1n‖vi‖L2(D×Yi−1,H~#(curl,Yi))+∑i=1n‖vi‖L2(D×Yi−1,H#1(Yi)/R). We then have the following result. Proposition 2.4 We define $$\boldsymbol{u}=(u_0,\{u_i\}, \{\frak u_i\})\in{\bf V}$$. Then $$\boldsymbol{u}$$ satisfies   B(u,v):=∫D∫Y[a(x,y)(curlu0+∑i=1ncurlyiui)⋅(curlv0+∑i=1ncurlyivi)  +b(x,y) (u0+∑i=1n∇yiui)⋅(v0+∑i=1n∇yivi)]dydx=∫Df(x)⋅v0(x)dx (2.10) for all $$\boldsymbol{v}=(v_0,\{v_i\},\{\frak v_i\})\in {\bf V}$$. Proof. Let $$v_0\in C^\infty_0(D)^3$$, $$v_i\in C^\infty_0(D, C^\infty_\#(Y_1,\ldots,C^\infty_\#(Y_i),\ldots))^3$$ and $${\frak v_i}\in C^\infty_0(D,C^\infty_\#(Y_1,\ldots,C^\infty_\# (Y_i),\ldots))$$ for $$i=1,\ldots,n$$. Let the test function $$v$$ in (2.5) be   v(x)=v0(x)+∑i=1nεi(vi(x,xε1,…,xεi)+∇vi(x,xε1,…,xεi)). We have    ∫D[aε(x)curluε(x)⋅(curlv0(x)+∑i=1nεicurlxvi(x,xε1,…,xεi)+∑i=1n∑j=1iεiεjcurlyjvi(x,xε1,…,xεi)   +  ∑i=1nεicurl∇vi(x,xε1,…,xεi))  +bε(x)uε(x)⋅(v0(x)+∑i=1nεivi(x,xε1,…,xεi)   +∑i=1nεi∇xvi(x,xε1,…,xεi)+∑i=1n∑j=1iεiεj∇yjvi(x,xε1,…,xεi))]dx  =∫Df(x)⋅(v0(x)+∑i=1nεivi(x,xε1,…,xεi)  +∑i=1nεi∇xvi(x,xε1,…,xεn)+∑i=1n∑j=1iεiεj∇yjvi(x,xε1,…,xεi)). Using multiscale convergence and the scale separation (2.2), letting $$\varepsilon$$ go to 0, we have    ∫D∫Y[a(x,y)(curlu0+∑i=1ncurlyiui)⋅(curlv0+∑i=1ncurlyivi)  +b(x,y)(u0+∑i=1n∇yiui)⋅(v0+∑i=1n∇yivi)]dydx =∫Df(x)⋅v0(x)dx+∫D∫Yf(x)⋅∑i=1n∇yivi(x,y1,…,yi)dydx =∫Df(x)⋅v0(x)dx. Using a density argument, we have (2.10). □ Proposition 2.5 The bilinear form $$B:{\bf V}\times{\bf V}\to \mathbb{R}$$ is coercive and bounded, i.e., there are positive constants $$C^*$$ and $$C_*$$ so that   B(u,v)≤C∗|||u||||||v|||andC∗|||u||||||u|||≤B(u,u) (2.11) for all $$\boldsymbol{u},\boldsymbol{v}\in {\bf V}$$. Problem (2.10) thus has a unique solution. The convergence relations (2.8) and (2.9) hold for the whole sequence $$\{u^{\varepsilon}\}_\varepsilon$$. Proof. It is easy to see that there is a positive constant $$C^*$$ such that   B(u,v)≤C∗|||u||||||v|||. Now we show that $$B$$ is coercive. We have from (2.1),   B(u,u) ≥c∗∫D∫Y(|curlu0+∑i=1ncurlyiui|2+|u0+∑i=1n∇yiui|2)dydx ≥c∫D∫Y(|curlu0|2+∑i=1n|curlyiui|2+|u0|2+|∇yiui|2)dydx≥c|||u|||2. We then get the conclusion from Lax–Milgram lemma. □ 3. FE discretization Let $$D$$ be a polygonal domain in $$\mathbb{R}^3$$. We consider a hierarchy of simplices $$\mathcal{T}^l$$ ($$l=0,1,\ldots$$), where $$\mathcal{T}^{l+1}$$ is obtained from $$\mathcal{T}^l$$ by dividing each simplex in $$\mathcal{T}^l$$ into eight tedrahedra. The mesh size of $$\mathcal{T}^l$$ is $$h_l={\mathcal O}(2^{-l})$$. For each tedrahedron $$T$$, we consider the edge FE space   R(T)={v:  v=α+β×x,  α,β∈R3}. When $$d=2$$, $$\mathcal{T}^{l+1}$$ is obtained from $$\mathcal{T}^l$$ by dividing each simplex in $$\mathcal{T}^l$$ into four congruent triangles. For each triangle $$T$$, we consider the edge FE space   R(T)={v:  v=(α1α2)+β(x2−x1)}, where $$\alpha_1,\alpha_2$$ and $$\beta$$ are constants. Alternatively, if the domain can be partitioned into a set of cubes, we can use edge FE on a cubic mesh instead (see Monk, 2003). We denote by $$\mathcal{P}_1(T)$$ the set of linear polynomials in each simplex $$T$$. In the following, we present the analysis for the three-dimensional case only; the two-dimensional case is similar. For the cube $$Y$$, we partition it into a hierarchy of simplices $$\mathcal{T}^l_\#,$$ which are distributed periodically. We consider the FE spaces   Wl ={v∈H0(curl,D), v|T∈R(T) ∀T∈Tl},Vl ={v∈H1(D), v|T∈P1(T) ∀T∈Tl},W#l ={v∈H#(curl,Y), v|T∈R(T) ∀T∈T#l} and   V#l={v∈H#1(Y), v|T∈P1(T) ∀T∈T#l}. For $$d=2,3$$, we have the following estimates (see Ciarlet, 1978; Monk, 2003):   infvl∈Wl‖v−vl‖H(curl,D)≤chls(‖v‖Hs(D)d+‖curlv‖Hs(D)d) for all $$v\in H_0({\rm curl\,}, D)\bigcap H^s({\rm curl\,},D)$$;   infvl∈W#l‖v−vl‖H#(curl,Y)≤chls(‖v‖Hs(Y)d+‖curlv‖Hs(Y)d) for all $$v\in H_\#({\rm curl\,}, Y)\bigcap H^s({\rm curl\,}, Y)$$;   infvl∈Vl‖v−vl‖L2(D)≤chls‖v‖Hs(D) for all $$v\in H^s(D)$$;   infvl∈V#l‖v−vl‖L2(Y)≤chls‖v‖Hs(Y) for all $$v\in H^s_\#(Y)$$ and   infvl∈V#l‖v−vl‖H#1(Y)≤chls‖v‖H1+s(Y) for all $$v\in H^1_\#(Y)\bigcap H^{1+s}(Y)$$. 3.1 Full tensor product FEs As $$L^2(D\times{\bf Y}_{i-1},\tilde H_\#({\rm curl\,},Y_i))\cong L^2(D)\otimes L^2(Y_1)\otimes\cdots\otimes L^2(Y_{i-1})\otimes \tilde H_\#({\rm curl\,},Y_i),$$ we use the tensor product FE space   Wil=Vl⊗V#l⊗⋯⊗V#l⏟i−1 times ⊗W#l to approximate $$u_i$$. Similarly, as $${\frak u_i}\in L^2(D\times{\bf Y}_{i-1},H^1_\#(Y))$$, we use the FE space   Vil=Vl⊗V#l⊗⋯⊗V#l⏟i times  to approximate $${\frak u_i}$$. We define the space   Vl=Wl×W1l×⋯×Wnl×V1l×⋯×Vnl. The full tensor product FE approximating problem is, find $$\boldsymbol{u}^L\in{\bf V}^L$$ so that   B(uL,vL)=∫Df(x)⋅v0L(x)dx  ∀vL=(v0L,{viL},viL)∈VL. (3.1) To get an error estimate for this FE approximating problem, we define the following regularity spaces for $${\frak u_i}$$ and $$u_i$$. For the functions $$u_i$$, we define the regularity space $$\mathcal{H}_i$$ of functions $$w$$ in $$L^2(D\times Y_1\times\cdots\times Y_{i-1},H^1_\#({\rm curl\,},Y_i))$$ such that for all $$k=1,2,3$$,   ∂w∂xk∈L2(D×Y1×⋯×Yi−1,H~#(curl,Yi)) and for all $$j=1,\ldots,i-1$$ and $$k=1,2,3$$,   ∂w∂(yj)k∈L2(D×Y1×⋯×Yi−1,H~#(curl,Yi)). In other words, for all $$w\in \mathcal{H}_i$$, $$w$$ belongs to $$L^2(D\times Y_1\times\cdots\times Y_{i-1},H^1_\#({\rm curl\,},Y_i))$$, $$L^2(Y_1\times\cdots\times Y_{i-1},H^1(D,\tilde H_\#({\rm curl\,},Y_i)))$$ and $$L^2(D\times\prod_{k<i,k\ne j}Y_k,H^1_\#(Y_j,\tilde H_\#({\rm curl\,},Y_i)))$$ for $$j=1,\ldots,i-1$$. For $$0<s<1$$, we define the space $$\mathcal{H}^s_i$$ by interpolation. It consists of functions $$w$$ such that $$w$$ belongs to $$L^2(D\times Y_1\times\cdots\times Y_{i-1},H^s_\#({\rm curl\,},Y_i))$$, $$L^2(Y_1\times\cdots\times Y_{i-1},H^s(D,\tilde H_\#({\rm curl\,},Y_i)))$$ and $$L^2(D\times\prod_{k<i,k\ne j}Y_k,H^s_\#(Y_j,\tilde H_\#({\rm curl\,},Y_i)))$$. We equip $$\mathcal{H}_i^s$$ with the norm   ‖w‖His =‖w‖L2(D×Y1×⋯×Yi−1,H#s(curl,Yi))+‖w‖L2(Y1×⋯×Yi−1,Hs(D,H~#(curl,Yi)))  +∑j=1i−1‖w‖L2(D×∏k<i,k≠j,H#s(Yj,H~#(curl,Yi))). We then have the following lemma. Lemma 3.1 For $$w\in \mathcal{H}_i^s$$,   infwl∈Wil‖w−wl‖L2(D×Y1×⋯×Yi−1,H~#(curl,Yi))≤chls‖w‖His. The proof of this lemma is similar to that for full tensor product FEs in Hoang & Schwab (2004/05) and Bungartz & Griebel (2004), using orthogonal projection. We refer to Hoang & Schwab (2004/05) and Bungartz & Griebel (2004) for details. We define $${\frak H_i}^s$$ as the space of functions $$w\in L^2(D\times Y_1\times\cdots\times Y_{i-1},H^{1+s}_\#(Y_i))$$ such that $$w\in L^2(Y_1\times\cdots\times Y_{i-1},H^s(D,H^1_\#(Y_i)))$$ and for all $$j=1,\ldots,i-1$$, $$ w\in L^2(D\times\prod_{k<i,k\ne j}Y_k,H^s_\#(Y_{j},H^1_\#(Y_i)))$$. We then define the norm   ‖w‖His =‖w‖L2(D×Y1×⋯×Yi−1,H#1+s(Yi))+‖w‖L2(Y1×⋯×Yi−1,Hs(D,H1(Yi)))  +∑j=1i−1‖w‖L2(D×∏k<i,k≠jYk,Hs(Yj,H1(Yi))). We have the following result. Lemma 3.2 For $$w\in {\frak H_i}^s$$,   infwl∈Vil‖w−wl‖L2(D×Y1×⋯×Yi−1,H#1(Yi))≤chls‖w‖His. We then define the regularity space   Hs=Hs(curl,D)×H1s×⋯×Hns×H1s×⋯×Hns with the norm   ‖w‖Hs=‖w0‖Hs(curl,D)+∑i=1n‖wi‖His+∑i=1n‖wi‖His for $$\boldsymbol{w}=(w_0,\{w_i\},\{\frak w_i\})\in \boldsymbol{\mathcal{H}}^s$$. We have the following approximation result. Lemma 3.3 For $$\boldsymbol{w}\in \boldsymbol{\mathcal{H}}^s$$  infwl∈Vl‖w−wl‖V≤chls‖w‖Hs. From the boundedness and coerciveness conditions (2.11), using Cea’s lemma, we deduce the following result. Proposition 3.4 If $$\boldsymbol{u}\in \boldsymbol{\mathcal{H}}^s$$, for the full tensor product FE approximating problem (3.1) we have the error estimate   ‖u−uL‖V≤chLs‖u‖Hs. (3.2) 3.2 Sparse tensor product FEs We define the following orthogonal projection:   Pl0:L2(D)→Vl,P#l0:L2(Y)→V#l with the convention $$P^{-10}=0$$, $$P^{-10}_\#=0$$. We define the following detail spaces:   Vl=(Pl0−P(l−1)0)Vl,  V#l=(P#l0−P#(l−1)0)Vl. Since   Vl=⨁0≤i≤lViandV#l=⨁0≤i≤lV#i, the full tensor product spaces $$W_i^L$$ and $$V_i^L$$ are defined as   WiL=(⨁0≤l0,…,li−1≤LVl0⊗V#l1⊗⋯⊗V#li−1)⊗W#L and   ViL=(⨁0≤l0,…,li−1≤LVl0⊗V#l1⊗⋯⊗V#li−1)⊗V#L. We then define the sparse tensor product FE spaces as   W^iL=⨁l0+⋯+li−1≤LVl0⊗V#l1⊗⋯⊗V#li−1⊗W#L−(l0+⋯+li−1) and   V^iL=⨁l0+⋯+li−1≤LVl0⊗V#l1⊗⋯⊗V#li−1⊗V#L−(l0+⋯+li−1). The function $$\boldsymbol{u}$$ is approximated by the space   V^L=WL⊗W^1L⊗⋯⊗W^nL⊗V^1L⊗⋯⊗V^nL. The sparse tensor product FE approximating problem is, find $$\widehat {\bf{u}}^L\in \hat{\bf V}^L$$ such that   B(u^L,v^L)=∫Df(x)⋅v^0L(x)dx  ∀v^L=(v^0L,{v^iL},{v^iL})∈V^L. (3.3) From the coerciveness and boundedness conditions in (2.11), using Cea’s lemma we deduce the error estimate for the sparse tensor product approximating problem   ‖u−u^L‖V≤cinfv^L∈VL‖u−v^L‖V. To quantify the error estimate, we use the following regularity spaces. We define $$\hat{\mathcal{H}}_i$$ as the space of functions $$w\in L^2(D\times Y_1\times\cdots\times Y_{i-1},H^1_\#({\rm curl\,},Y_i)),$$ which are periodic with respect to $$y_j$$ with the period being $$Y_j$$ ($$j=1,\ldots,i-1$$) such that for any $$\alpha_0,\alpha_1,\ldots,\alpha_{i-1}\in \mathbb{N}_0^d$$ with $$|\alpha_k|\le 1$$ for $$k=0,\ldots,i-1$$,   ∂|α0|+|α1|+⋯+|αi−1|∂xα0∂y1α1⋯∂yi−1αi−1w∈L2(D×Y1×⋯×Yi−1,H#1(curl,Yi)). We equip $$\hat{\mathcal{H}}_i$$ with the norm   ‖w‖H^i=∑αj∈Rd,|αj|≤10≤j≤i−1‖∂|α0|+|α1|+⋯+|αi−1|∂xα0∂y1α1⋯∂yi−1αi−1w‖L2(D×Y1×⋯×Yi−1,H#1(curl,Yi)). We can write $$\hat{\mathcal{H}}_i$$ as $$H^1(D,H^1_\#(Y_1,\ldots,H^1_\#(Y_{i-1},H^1_\#({\rm curl\,},Y_i)),\ldots))$$. By interpolation, we define $$\hat{\mathcal{H}}_i^s=H^s(D,H^s_\#(Y_1,\ldots,H^s_\#(Y_{i-1},H^s_\#({\rm curl\,},Y_i)),\ldots))$$ for $$0<s<1$$. We define $$\hat{\frak H}_i$$ as the space of functions $$w\in L^2(D\times Y_1\times\cdots\times Y_{i-1},H^2_\#(Y_i))$$ that are periodic with respect to $$y_j$$ with the period being $$Y_j$$ for $$j=1,\ldots,i-1$$ such that $$\alpha_0,\alpha_1,\ldots,\alpha_{i-1}\in \mathbb{N}_0^d$$ with $$|\alpha_k|\le 1$$ for $$k=0,\ldots,i-1$$,   ∂|α0|+|α1|+⋯+|αi−1|∂xα0∂y1α1⋯∂yi−1αi−1w∈L2(D×Y1×⋯×Yi−1,H#2(Yi)). The space $$\hat{\frak H}_i$$ is equipped with the norm   ‖w‖H^i=∑αj∈Rd,|αj|≤10≤j≤i−1‖∂|α0|+|α1|+⋯+|αi−1|∂xα0∂y1α1⋯∂yi−1αi−1w‖L2(D×Y1×⋯×Yi−1,H#2(Yi)). We can write $$\hat{\frak H}_i$$ as $$H^1(D,H^1_\#(Y_1,\ldots,H^1_\#(Y_{i-1},H^2_\#(Y_i))))$$. By interpolation, we define the space $$\hat{\frak H}_i^s:=H^s(D,H^s(Y_1,\ldots,H^s(Y_{i-1},H^{1+s}_\#(Y_i))))$$. The regularity space $$\hat{\boldsymbol{\mathcal{H}}}^s$$ is defined as   H^s=Hs(curl,D)×H^1s×⋯H^ns×H^1s×⋯×H^ns. Lemmas 3.5 and 3.6 present the approximating properties of functions in $$\hat{\mathcal{H}}_i^s$$ and $${\hat{\frak H}_i^s}$$. The proofs follow from those for sparse tensor products in Hoang & Schwab (2004/05) and Bungartz & Griebel (2004). Lemma 3.5 For $$w\in \hat{\mathcal{H}}_i^s$$,   infwL∈W^iL‖w−wL‖L2(D×Y1×⋯×Yi−1,H#(curl,Yi))≤cLi/2hLs‖w‖H^is. Similarly we have the following lemma. Lemma 3.6 For $$w\in \hat{\frak H}_i^s$$,   infwL∈V^iL‖w−wL‖L2(D×Y1×⋯×Yi−1,H#1(Yi))≤cLi/2hLs‖w‖H^is. From these lemmas we deduce the following result. Lemma 3.7 For $$\boldsymbol{w}\in \hat{\boldsymbol{\mathcal{H}}}^s$$,   infwL∈V^L‖w−wL‖V≤cLn/2hLs‖w‖H^s. From this we deduce the following error estimate for the sparse tensor product FE problem (3.3). Proposition 3.8 If the solution $$\boldsymbol{u}$$ of problem (2.10) belongs to $$\hat{\boldsymbol{\mathcal{H}}}^s$$ then   ‖u−u^‖V≤cLn/2hLs‖u‖H^s. Remark 3.9 The dimension of the full tensor product FE space $${\bf V}^L$$ is $${\mathcal O}(2^{dnL}),$$ which is very large when $$L$$ is large. The dimension of the sparse tensor product FE space $$\hat{\bf V}^L$$ is $${\mathcal O}(L^n2^{dL}),$$ which is essentially equal to the number of degrees of freedom for solving a problem in $$\mathbb{R}^d$$ obtaining the same level of accuracy. 4. Convergence in physical variables We employ the FE solutions for the multiscale homogenized Maxwell-type equation (2.10) in the previous section to derive numerical correctors for the solution $$u^{\varepsilon}$$ of the multiscale problem (2.4). In the two-scale case, we derive the homogenization error explicitly in terms of $$\varepsilon$$ so that an error in terms of the microscopic scale $$\varepsilon$$ and the mesh size is obtained for the numerical corrector. We consider the general case, where the solution $$u^0$$ of the homogenized problem belongs to the space $$H^s({\rm curl\,},D)$$ for $$0<s\le 1$$, thus generalizing the standard homogenization rate of convergence $$\varepsilon^{1/2}$$ for elliptic problems (see e.g, Bensoussan et al., 1978; Jikov et al., 1994). This is a new result in homogenization theory. We present it for two-scale Maxwell-type equations, but the procedure works verbatim for two-scale elliptic and elasticity problems, where the solutions of the homogenized problems belong to $$H^{1+s}(D)$$. We present this section for the case $$d=3$$; the case $$d=2$$ is similar. 4.1 Two-scale problems For the two-scale case, we denote the function $$a(x,\boldsymbol{y})$$ by $$a(x,y)$$. The two-scale homogenized equation becomes    ∫D∫Y[a(x,y)(curlu0+curlyu1)⋅(curlv0+curlyv1)+b(x,y)(u0+∇yu1)⋅(v0+∇yv1)]dydx  =∫Df(x)⋅v0(x)dx. We first let $$v_0=0$$, $$v_1=0$$ and deduce that   ∫D∫Yb(x,y)(u0+∇yu1)⋅∇yv1dydx=0. For each $$r=1,2,3$$, let $$w^r(x,\cdot)\in L^2(D, H^1_\#(Y)/\mathbb{R})$$ be the solution of the problem   ∫D∫Yb(x,y)(er+∇ywr)⋅∇yψdydx=0  ∀ψ∈L2(D,H#1(Y)/R), (4.1) where $$e_r$$ is the vector in $$\mathbb{R}^3$$ with all the components being 0, except the $$r$$th component, which equals 1. This is the standard cell problem in elliptic homogenization. From this we have   u1(x,y)=wr(x,y)u0r(x). (4.2) Therefore,   ∫D∫Yb(x,y)(u0+∇yu1)⋅v0dxdy=∫Db0(x)u0(x)⋅v0(x)dx, where the positive-definite matrix $$b^0(x)$$ is defined as   bij0(x)=∫Yb(x,y)(ej+∇wj(x,y))⋅(ei+∇ywi(x,y))dy, (4.3) which is the usual homogenized coefficient for elliptic problems with the two-scale coefficient matrix $$b^\varepsilon$$. Let $$v_0=0$$ and $${\frak v_1}=0$$. We have   ∫D∫Ya(x,y)(curlu0+curlyu1)⋅curlyv1dydx=0 for all $$v_1\in L^2(D, \tilde H_\#({\rm curl\,},Y))$$. For each $$r=1,2,3$$, let $$N^r\in L^2(D,\tilde H_\#({\rm curl\,},Y))$$ be the solution of   ∫D∫Ya(x,y)(er+curlyNr)⋅curlyvdydx=0 (4.4) for all $$v\in L^2(D, \tilde H_\#({\rm curl\,},Y))$$. We have   u1=(curlu0(x))rNr(x,y). (4.5) The homogenized coefficient $$a^0$$ is determined by   aij0(x)=∫Ya(x,y)ip(ejp+(curlyNj)p)dy=∫Ya(x,y)(ej+curlyNj)⋅(ei+curlyNi)dy. (4.6) We have   ∫D∫Ya(x,y)(curlu0+curlyu1)⋅curlv0dxdy=∫Da0(x)curlu0(x)⋅curlv0(x)dx. The homogenized problem is   ∫D[a0(x)curlu0(x)⋅curlv0(x)+b0(x)u0(x)⋅v0(x)]dx=∫Df(x)⋅v0(x)dx  ∀v0∈H0(curl,D). (4.7) Following the procedure for deriving the homogenization error (Bensoussan et al., 1978; Jikov et al., 1994), we have the following homogenization error estimate. Theorem 4.1 Assume that $$a\in C(\bar D, C(\bar Y))^{3\times 3}$$, $$u_0\in H^1({\rm curl\,};D)$$, $$N^r\in C^1(\bar D,C(\bar Y))^3$$, $${\rm curl}_yN^r\in C^1(\bar D,C(\bar Y))^3$$, $$w^r\in C^1(\bar D,C^1(\bar Y))$$ for all $$r=1,2,3$$, then2  ‖uε−[u0+∇yu1(⋅,⋅ε)]‖L2(D)3≤cε1/2 and   ‖curluε−[curlu0+curlyu1(⋅,⋅ε)]‖L2(D)3≤cε1/2. The proof of this theorem uses the functions $$G_r$$ and $$g_r$$ defined in (A.2) and (A.3) below. For $$u_0\in H^s({\rm curl\,},D)$$ when $$0<s<1$$, we have the following homogenization error estimate. Theorem 4.2 Assume that $$a\in C(\bar D,C(\bar Y))^{3\times 3}$$, $$u_0\in H^s({\rm curl\,},D)$$, $$N^r\in C^1(\bar D, C(\bar Y))^3$$, $${\rm curl}_yN^r\in C^1(\bar D,C(\bar Y))^3$$ and $$w^r\in C^1(\bar D,C^1(\bar Y))$$ for all $$r=1,2,3$$, then   ‖uε−[u0+∇yu1(⋅,⋅ε)‖L2(D)3≤cεs/(1+s) and   ‖curluε−[curlu0+curlyu1(⋅,⋅ε)]‖L2(D)3≤cεs/(1+s). We present the proof of this theorem in Appendix A. To employ the FE solutions to construct numerical correctors for $$u^{\varepsilon}$$, we define the following operator:   Uε(Φ)(x)=∫YΦ(ε[xε]+εz,{xε})dz. (4.8) Let $$D^\varepsilon$$ be a $$2\varepsilon$$ neighbourhood of $$D$$. Regarding $${\it{\Phi}}$$ as zero when $$x$$ is outside $$D$$, we have   ∫DεUε(Φ)(x)dx=∫D∫YΦ(x,y)dxdy. (4.9) The proof of (4.9) may be found in Cioranescu et al. (2008). We have the following result. Lemma 4.3 Assume that for $$r=1,2,3$$, $${\rm curl}_yN^r(x,y)\in C^1(\bar D, C(\bar Y))^3$$ and $$u_0\in H^s({\rm curl\,},D)$$, then   ‖curlyu1(⋅,⋅ε)−Uε(curlyu1)‖L2(D)3≤cεs. We prove this lemma in Appendix B. We then have the following result. Theorem 4.4 Assume that $$a\in C(\bar D,C(\bar Y))^{3\times 3}$$, $$u_0\in H^s({\rm curl\,},D)$$, $$N^r\in C^1(\bar D,C(\bar Y))^3$$, $${\rm curl}_yN^r\in C^1(\bar D,C(\bar Y))^3$$ and $$w^r\in C^1(\bar D,C^1(\bar Y))$$ for all $$r=1,2,3$$. Then for the full tensor product FE solution $$(u_0^L,u_1^L,{\frak u_1}^L)$$ we have   ‖uε−u0L−Uε(∇uu1L)‖L2(D)3≤c(εs/(1+s)+hLs) and   ‖curluε−curlu0L−Uε(curlyu1L)‖L2(D)3≤c(εs/(1+s)+hLs). Proof. From Lemma 4.3, we have    ‖curluε−curlu0L−Uε(curlyu1L)‖L2(D)3  ≤‖curluε−curlu0−curlyu1(⋅,⋅ε)‖L2(D)3  +‖curlu0−curlu0L‖L2(D)3+‖curlyu1(⋅,⋅ε)−Uε(curlyu1)‖L2(D)3  +‖Uε(curlyu1)−Uε(curlyu1L)‖L2(D)3. Using the fact that $$(\mathcal{U}^\varepsilon({\it{\Phi}}))^2\le \mathcal{U}({\it{\Phi}}^2)$$ and (4.9), we have   ‖Uε(curlyu1)−Uε(curlyu1L)‖L2(D)3≤‖curlyu1−curlyu1L‖L2(D×Y)3≤chLs. This together with (3.2), Theorem 4.2 and Lemma 4.3 gives   ‖curluε−curlu0L−Uε(curlyu1L)‖L2(D)3≤c(εs/(1+s)+hLs). Similarly, we have   ‖uε−u0L−Uε(∇uu1L)‖L2(D)3≤c(εs/(1+s)+hLs). □ For the sparse tensor product FE approximation, we have the following result. Theorem 4.5 Assume that $$a\in C(\bar D,C(\bar Y))^{3\times 3}$$, $$u_0\in H^s({\rm curl\,},D)$$, $$N^r\in C^1(\bar D,C(\bar Y))^3$$, $${\rm curl}_yN^r\in C^1(\bar D,C(\bar Y))^3$$ and $$w^r\in C^1(\bar D,C^1(\bar Y))$$ for all $$r=1,2,3$$. Then for the sparse tensor product FE solution $$(\hat u_0^L,\hat u_1^L,\hat{\frak u}_1^L)$$ we have   ‖uε−u^0L−Uε(∇yu^1L)‖L2(D)3≤c(εs/(1+s)+Ln/2hLs) and   ‖curluε−curlu^0L−Uε(curlyu^1L)‖L2(D)3≤c(εs/(1+s)+Ln/2hLs). 4.2 Multiscale problems For multiscale problems, we do not have an explicit homogenization rate of convergence. However, for the case where $$\varepsilon_i/\varepsilon_{i+1}$$ is an integer for all $$i=1,\ldots,n-1$$ we can derive a corrector for the solution $$u^{\varepsilon}$$ of the multiscale problem from the FE solutions of the multiscale homogenized problem. For each function $$\phi\in L^1(D),$$ which is understood as 0 outside $$D$$, we define a function in $$L^1(D\times{\bf Y})$$:   Tnε(ϕ)(x,y)=ϕ(ε1[xε1]+ε2[y1ε2/ε1]+⋯+εn[yn−1εn/εn−1]+εnyn). Letting $$D^{\varepsilon_1}$$ be the $$2\varepsilon_1$$ neighbourhood of $$D$$, we have   ∫Dϕdx=∫Dε1∫Y1⋯∫YnTnε(ϕ)dyn⋯dy1dx (4.10) for all $$\phi \in L^1(D)$$. If a sequence $$\{\phi^\varepsilon\}_\varepsilon$$$$(n+1)$$-scale converges to $$\phi(x,y_1,\ldots,y_n)$$ then   Tnε(ϕ)⇀ϕ(x,y1,…,yn) in $$L^2(D\times Y_1\times\ldots\times Y_n)$$. Thus, when $$\varepsilon\to 0$$,   Tnε(curluε)⇀curlu0+curly1u1+⋯+curlynun (4.11) and   Tnε(uε)⇀u0+∇y1u1+⋯+∇ynun (4.12) in $$L^2(D\times{\bf Y})^3$$. To deduce an approximation of $$u^{\varepsilon}$$ in $$H({\rm curl\,},D)$$ in terms of the FE solution, we use the operator $$\mathcal{U}_n^\varepsilon$$ , which is defined as   Unε(Φ)(x) =∫Y1⋯∫YnΦ(ε1[xε1]+ε1t1,ε2ε1[ε1ε2{xε1}]   +ε2ε1t2,⋯, εnεn−1[εn−1εn{xεn−1}]+εnεn−1tn,{xεn})dtn⋯dt1 for all functions $${\it{\Phi}}\in L^1(D\times{\bf Y})$$. For each function $${\it{\Phi}}\in L^1(D\times{\bf Y})$$ we have   ∫Dε1Unε(Φ)dx=∫D∫YΦ(x,y)dydx. (4.13) The proofs for these facts may be found in Cioranescu et al. (2008). We then have the following corrector result. Proposition 4.6 The solution $$u^{\varepsilon}$$ of problem (2.5) and the solution $$(u_0,\{u_i\}\,\{{\frak u_i}\})$$ of problem (2.10) satisfies   limε→0‖uε−[u0+Unε(∇y1u1)+⋯+Unε(∇ynun)‖L2(D)3=0 (4.14) and   limε→0‖curluε−[curlu0+Unε(curly1u1)+⋯+Unε(curlynun)]‖L2(D)3=0. (4.15) Proof. We consider the expression    ∫D∫Y[Tnε(aε)(Tnε(curluε)−(curlu0+curly1u1+⋯+curlynun)) ⋅(Tnε(curluε)−(curlu0+curly1u1+⋯+curlynun)) +Tnε(bε)(Tnε(uε)−(u0+∇y1u1+⋯+∇ynun))⋅(Tnε(uε)−(u0+∇y1u1+⋯+∇ynun))dydx. Using (2.5), (2.10), (4.10), (4.11) and (4.12), we deduce that this expression converges to 0. From (2.1) we have   limε→0‖Tnε(curluε)−(curlu0+curly1u1+⋯+curlynun)‖L2(D×Y1×…×Yn)3=0 and   limε→0‖Tnε(uε)−(u0+∇y1u1+⋯+∇ynun)‖L2(D×Y1×⋯×Yn)3=0. From (4.13) and the fact that $$\mathcal{U}^\varepsilon_n({\it{\Phi}})^2\le \mathcal{U}^\varepsilon_n({\it{\Phi}}^2)$$, we have    ∫D|Unε(Tnε(curluε)−(curlu0+curly1u1+⋯+curlynun)(x)|2dx ≤∫DUnε(|Tnε(curluε)−(curlu0+curly1u1+⋯+curlynun|2)(x)|dx ≤∫D∫Y|Tnε(curluε)−(curlu0+curly1u1+⋯+curlynun|2)dydx, which converges to 0 when $$\varepsilon\to 0$$. Using $$\mathcal{U}^\varepsilon_n(\mathcal{T}^\varepsilon_n({\it{\Phi}}))={\it{\Phi}}$$, we get (4.15). We derive (4.14) similarly. □ We then deduce the numerical corrector result. Theorem 4.7 For the full tensor product FE approximation solution $$\boldsymbol{u}^L=(u_0^L,\{u_i^L\},\{{\frak u_i}^L\})$$ in (3.1), we have   limε→0L→∞‖uε−[u0L+Unε(∇y1u1L)+⋯+Unε(∇ynunL)]‖L2(D)3=0 (4.16) and   limε→0L→∞‖curluε−[curlu0L+Unε(curly1u1L)+⋯+Unε(curlynunL)]‖L2(D)3=0. (4.17) Proof. We note that    ‖Unε(curly1u1+⋯+curlynun)−Unε(curly1u1L+⋯+curlynunL)‖L2(D)3 ≤∫DUnε(|(curly1u1+⋯+curlynun)−(curly1u1L+⋯+curlynunL)|2)(x)dx ≤∫D∫Y|(curly1u1+⋯+curlynun)−(curly1u1L+⋯+curlynunL)|2dydx, which converges to 0 when $$L\to \infty$$. From this and (4.15), we get (4.17). We obtain (4.16) in the same way. □ Remark 4.8 As $${\bf V}^{\lceil L/n\rceil}\subset\hat{\bf V}^L$$, the result in Theorem 4.7 also holds for the sparse tensor product FE solution $$\widehat {\bf{u}}^L$$. Since we do not have an explicit homogenization error for problems with more than two scales, we do not distinguish the two cases of full and sparse tensor FE approximations. 5. Regularity of $$\boldsymbol{N^r}$$, $$\boldsymbol{w^r}$$ and $$\boldsymbol{u_0}$$ We show in this section that the regularity requirements for obtaining the sparse tensor product FE error estimate and the homogenization error estimate in the previous sections are achievable. We present the results for the two-scale case in detail. The multiscale case is similar; we summarize it in Remark 5.7. We first prove the following lemma. Lemma 5.1 Let $$\psi\in H_\#({\rm curl\,},Y)\bigcap H_\#({\rm div},Y)$$. Assume further that $$\int_Y\psi(y)\,{\rm d}y=0$$. Then $$\psi\in H^1_\#(Y)^3$$ and   ‖ψ‖H1(Y)3≤c(‖curlyψ‖L2(Y)3+‖divyψ‖L2(Y)). Proof. Let $$\omega\subset\mathbb{R}^3$$ be a smooth domain such that $$\omega\supset Y$$. Let $$\eta\in \mathcal{D}(\omega)$$ be such that $$\eta(y)=1$$ when $$y\in Y$$. We have   curly(ηψ)=ηcurlyψ+∇yη×ψ∈L2(ω)3 and   divy(ηψ)=∇yη⋅ψ+ηdivyψ∈L2(ω)3. Together with the zero boundary condition, we conclude that $$\eta\psi\in H^1(\omega)^3$$ so $$\psi\in H^1(Y)^3$$. We note that   ∫Y(divyψ(y)2+|curlyψ(y)|2)dy=∑i,j=13∫Y(∂ψi∂yj)2+∑i≠j∫Y∂ψi∂yi∂ψj∂yjdy−∑i≠j∫Y∂ψj∂yi∂ψi∂yjdy. Assume that $$\psi$$ is a smooth periodic function. We have   ∫Y∂ψi∂yi∂ψj∂yjdy=∫Y[∂∂yi(ψi∂ψj∂yj)−ψi∂2ψj∂yi∂yj]dy=−∫Yψi∂2ψj∂yi∂yjdy as $$\psi$$ is periodic. Similarly, we have   ∫Y∂ψi∂yj∂ψj∂yidy=∫Y[∂∂yj(ψi∂ψj∂yi)−ψi∂2ψj∂yj∂yi]dy=−∫Yψi∂2ψj∂yj∂yidy. Thus,   ∫Y∂ψi∂yi∂ψj∂yjdy=∫Y∂ψi∂yj∂ψj∂yidy. Therefore,   ‖∇yψ‖L2(Y)32=‖divyψ‖L2(Y)2+‖curlyψ‖L2(Y)32. Using a density argument, this holds for all $$\psi\in H^1_\#(Y)^3$$. As $$\int_Y\psi(y)\,{\rm d}y=0$$, from the Poincaré inequality we deduce   ‖ψ‖H1(Y)3≤c(‖divyψ‖L2(Y)+‖curlyψ‖L2(Y)3). □ Lemma 5.2 Let $$\alpha\in C^1_\#(\bar Y)^{3\times 3}$$ be uniformly bounded, positive definite and symmetric for all $$y\in \bar Y$$. Let $$F\in L^2(Y)$$, extending periodically to $$\mathbb{R}^3$$. Let $$\psi\in H^1_\#(Y)^3$$ satisfy the equation   curly(α(y)curlyψ(y))=F(y). Then $${\rm curl}_y\psi\in H^1_\#(Y)^3$$ and   ‖curlyψ‖H1(Y)3≤c(‖F‖L2(Y)3+‖ψ‖H1(Y)3). Proof. Let $$\omega\supset Y$$ be a smooth domain. Let $$\eta\in \mathcal{D}(\omega)$$ be such that $$\eta(y)=1$$ for $$y\in Y$$. We have   curly(αcurly(ηψ)) =curly(αηcurlyψ)+curly(α∇yη×ψ) =ηcurly(αcurlyψ)+∇yη×(αcurlyψ)+curly(α∇yη×ψ). Let $$U=\alpha{\rm curl}_y(\eta\psi)$$. We have   ‖curlyU‖L2(ω)3≤c(‖F‖L2(ω)+‖ψ‖H1(ω)3)≤c(‖F‖L2(Y)3+‖ψ‖H1(Y)3). Further,   ‖U‖L2(ω)3=‖α(∇yη×ψ+ηcurlyψ)‖|L2(ω)3≤c‖ψ‖H1(ω)3≤c‖ψ‖H1(Y)3. As $$\eta\in\mathcal{D}(\omega)$$, $$U$$ has compact support in $$\omega$$ so $$U$$ belongs to $$H_0({\rm curl\,},\omega)$$. Thus, we can write   U=z+∇Φ, where $$z\in H^1_0(\omega)^3$$ and $${\it{\Phi}}\in H^1_0(\omega)$$ satisfy   ‖z‖H1(ω)3≤c‖U‖H(curl,ω)  and  ‖Φ‖H1(ω)≤c‖U‖H(curl,ω). From $${\rm div}_y(\alpha^{-1}U)=0$$ we deduce that   divy(α−1∇Φ)=−divy(α−1z)∈L2(ω). Since $$\alpha\in C^1(\bar\omega)^{3\times 3}$$ and is uniformly bounded and positive definite, $$\alpha^{-1}\in C^1(\bar\omega)^{3\times 3}$$ and is uniformly positive definite. Therefore, $${\it{\Phi}}\in H^2(\omega)$$ and satisfies   ‖Φ‖H2(ω)≤c‖z‖H1(ω)3≤c‖U‖H(curl,ω). Thus, $$U\in H^1(\omega)^3$$ and $$\|U\|_{H^1(\omega)^3}\le c\|U\|_{H({\rm curl\,},\omega)}\le c(\|F\|_{L^2(Y)^3}+\|\psi\|_{H^1(Y)^3})$$. From $${\rm curl}_y(\eta\psi)=\alpha^{-1}U$$, we deduce that $${\rm curl}_y(\eta\psi)\in H^1(\omega)^3$$ so $${\rm curl}_y\psi\in H^1(Y)^3$$ and   ‖curlyψ‖H1(Y)3≤c(‖F‖L2(Y)3+‖ψ‖H1(Y)3). □ We then prove the following result on the regularity of $$N^r$$. Proposition 5.3 Assume that $$a(x,y)\in C^1(\bar D,C^2(\bar Y))^{3\times 3}$$; then $${\rm curl}_yN^r(x,y)\in C^1(\bar D,C(\bar Y))^3$$ and we can choose a version of $$N^r$$ in $$L^2(D,\tilde H_\#({\rm curl\,},Y))$$ so that $$N^r(x,y)\in C^1(\bar D, C(\bar Y))^3$$. Proof. We can choose a version of $$N^r$$ so that $${\rm div}_yN^r=0$$. Indeed, let $${\it{\Phi}}(x,\cdot)\in L^2(D, H^1_\#(Y))$$ be such that $$\Delta_y{\it{\Phi}}=-{\rm div}_yN^r$$; then $${\rm curl}_y(N^r+\nabla_y{\it{\Phi}})={\rm curl}_y N^r$$ and $${\rm div}_y(N^r+\nabla_y{\it{\Phi}})=0$$. Further we can choose $$N^r$$ so that $$\int_YN^r(x,y)\,{\rm d}y=0$$. From Lemma 5.1, we have   ‖Nr(x,⋅)‖H1(Y)3≤c‖curlyNr(x,⋅)‖L2(Y)3, which is uniformly bounded with respect to $$x$$. From (4.4) and Lemma 5.2, we deduce that   ‖curlyNr(x,⋅)‖H1(Y)3≤c‖curly(a(x,⋅)er‖L2(Y)3+‖Nr(x,⋅)‖H1(Y)3, which is uniformly bounded with respect to $$x$$. For each index $$q=1,2,3$$, we have that $${\rm curl}_y{\partial\over\partial y_q}N^r(x,\cdot)$$ is uniformly bounded in $$L^2(Y)$$ and $${\rm div}_y{\partial\over\partial y_q}N^r(x,\cdot)=0$$. Therefore, from Lemma 5.1, $${\partial\over\partial y_q}N^r(x,\cdot)$$ is uniformly bounded in $$H^1(Y)$$. We note that   ∂∂yq(curly(a(x,⋅)curlyNr))=−∂∂yqcurly(a(x,⋅)er)∈L2(Y). Thus,   curly(acurly∂Nr∂yq)=∂∂yq(curly(a(x,y)curlyNr))−curly(∂a∂yqcurlyNr)∈L2(Y). From Lemma 5.2 we deduce that $${\rm curl}_y{\partial N^r\over\partial y_q}(x,\cdot)$$ is uniformly bounded in $$H^1(Y)$$ so that $${\rm curl}_y N^r(x,\cdot)$$ is uniformly bounded in $$H^2(Y)\subset C(\bar Y)$$. We now show that $${\rm curl}_yN^r\in C^1(\bar D,H^2(Y))^3\subset C^1(\bar D,C(\bar Y))^3$$. Fix $$h\in\mathbb{R}^3$$. From (4.4) we have   curly(a(x,y)curly(Nr(x+h,y)−Nr(x,y))) =−curly((a(x+h,y)−a(x,y))er) −curly((a(x+h,y)−a(x,y))curlyNr(x+h,y)). The smoothness of $$a$$ and the uniform boundedness of $${\rm curl}_yN^r(x,\cdot)$$ in $$L^2(Y)^3$$ gives   limh→0‖curly(Nr(x+h,⋅)−Nr(x,⋅))‖L2(Y)3=0. (5.1) From Lemma 5.1 we have that $$N^r(x+h,\cdot)-N^r(x,\cdot)\in H^1(Y)^3$$ and   ‖Nr(x+h,⋅)−Nr(x,⋅)‖H1(Y)3≤c‖curly(Nr(x+h,⋅)−Nr(x,⋅))‖L2(Y)3, which converges to 0 when $$|h|\to 0$$. From Lemma 5.2, we have    ‖curly(Nr(x+h,⋅)−Nr(x,⋅))‖H1(Y)3 ≤‖−curly((a(x+h,⋅)−a(x,⋅))er)−curly((a(x+h,⋅)−a(x,⋅))curlyNr(x+h,⋅))‖L2(Y)3 +‖Nr(x+h,⋅)−Nr(x,⋅)‖H1(Y)3→0  when |h|→0. (5.2) We have further that   curly(a(x,y)curly∂∂yq(Nr(x+h,y)−Nr(x,y)))=−curly(∂a∂yq(x,y)curly(Nr(x+h,y)−Nr(x,y)))  −∂∂yqcurly((a(x+h,y)−a(x,y))er)−∂∂yqcurly((a(x+h,y)−a(x,y))curlyNr(x+h,y)). (5.3) From this we have   ‖curly∂∂yq(Nr(x+h,y)−Nr(x,y))‖L2(Y)3 ≤c‖curly(Nr(x+h,⋅)−Nr(x,⋅))‖L2(Y)3  +c‖a(x+h,⋅)−a(x,⋅)‖W1,∞(Y)3→0  when |h|→0, so from Lemma 5.1 we have   ‖∂∂yq(Nr(x+h,y)−Nr(x,y))‖H1(Y)3→0  when |h|→0. As the right-hand side of (5.3) converges to 0 in the $$L^2(Y)^3$$ norm when $$|h|\to 0$$, we deduce from Lemma 5.2 that   ‖curly∂∂yq(Nr(x+h,y)−Nr(x,y))‖H1(Y)3→0  when |h|→0. (5.4) We have   curly[a(x,y)curly(Nr(x+h,y)−Nr(x,y)h)] =−curly((a(x+h,y)−a(x,y)h)er)  −curly(a(x+h,y)−a(x,y)hcurlyNr(x+h,y)). Let $$\chi^r(x,\cdot)\in \tilde H_\#({\rm curl\,},Y)$$ with $${\rm div}_y\chi^r(x,)=0$$ be the solution of the problem   curly(a(x,y)curlyχr(x,⋅))=−curly(∂a∂xqer)−curly(∂a∂xqcurlyNr(x,y)). We deduce that   curly(a(x,y)curly(Nr(x+h,y)−Nr(x,y)h−χr(x,y))) =−curly((a(x+h,y)−a(x,y)h−∂a∂xq(x,y))er) −curly((a(x+h,y)−a(x,y)h−∂a∂xq(x,y))curlyNr(x+h,y)) −curly(∂a∂xq(x,y)curly(Nr(x+h,y)−Nr(x,y))):=I1. (5.5) Let $$h\in \mathbb{R}^3$$ be a vector with all components 0 except for the $$q$$th component. We have   ‖curly(Nr(x+h,⋅)−Nr(x,⋅)h−χr(x,⋅))‖L2(Y)3 ≤c‖a(x+h,⋅)−a(x,⋅)h−∂a∂xq(x,⋅)‖L∞(Y) +c‖curly(Nr(x+h,⋅)−Nr(x,⋅)‖L2(Y)3, (5.6) which converges to 0 when $$|h|\to 0$$ due to (5.1). Thus, we deduce from Lemma 5.1 that   lim|h|→0‖Nr(x+h,⋅)−Nr(x,⋅)h−χr(x,⋅)‖H1(Y)3=0. (5.7) From Lemma 5.2, we have    lim|h|→0‖curly(Nr(x+h,⋅)−Nr(x,⋅)h−χr(x,⋅))‖H1(Y)3 ≤lim|h|→0‖I1(x,⋅)‖L2(Y)3+‖Nr(x+h,⋅)−Nr(x,⋅)h−χr(x,⋅)‖H1(Y)3=0 (5.8) due to (5.2) and (5.7). Let $$p=1,2,3$$. We then have   curly(a(x,y)curly∂∂yp(Nr(x+h,y)−Nr(x,y)h−χr(x,y))) =−curly(∂a∂yp(x,y)curly(Nr(x+h,y)−Nr(x,y)h−χr(x,y))) −∂∂ypcurly((a(x+h,y)−a(x,y)h−∂a(x,y)∂xq)er) −∂∂ypcurly((a(x+h,y)−a(x,y)h−∂a∂xq(x,y))curlyNr(x+h,y)) −∂∂ypcurly(∂a∂xq(x,y)curly(Nr(x+h,y)−Nr(x,y))), which converges to 0 in $$L^2(Y)$$ for each $$x$$ due to (5.4), (5.8) and the uniform boundedness of $$\|{\rm curl\,} N^r(x,\cdot)\|_{H^2(Y)^3}$$. We have   ‖curly∂∂yp(Nr(x+h,⋅)−Nr(x,⋅)h−χr(x,⋅))‖L2(Y)3  ≤c‖curly(Nr(x+h,⋅)−Nr(x,⋅)h−χr(x,⋅))‖L2(Y)3  +c‖a(x+h,⋅)−a(x,⋅)h−∂a(x,⋅)∂xq‖W1,∞(Y)3+c‖curly(Nr(x+h,⋅)−Nr(x,⋅)‖H1(Y)3, which converges to 0 when $$|h|\to 0$$, so from Lemma 5.1,   lim|h|→0‖∂∂yp(Nr(x+h,⋅)−Nr(x,⋅)h−χr(x,⋅))‖H1(Y)3=0. (5.9) Therefore, $$N^r\in C^1(\bar D,H^2(Y))^3\subset C^1(\bar D,C(\bar Y))^3$$. We then get from Lemma 5.2 that   lim|h|→0‖curly∂∂yp(Nr(x+h,⋅)−Nr(x,⋅)h−χr(x,⋅))‖H1(Y)3=0. Thus, $${\rm curl}_yN^r\in C^1(\bar D,H^2(Y))^3\subset C^1(\bar D,C(\bar Y))^3$$. □ Proposition 5.4 Assume that $$b(x,y)\in C^1(\bar D,C^2(\bar Y))^{3\times 3}$$. The solution $$w^r$$ of cell problem (4.1) belongs to $$C^1(\bar D, C^1(\bar Y))$$. Proof. The cell problem (4.1) can be written as   −∇y⋅(b(x,y)∇ywr(x,y))=∇y(b(x,y)er). Fixing $$x\in \bar D$$, the right-hand side is bounded uniformly in $$H^1(Y)$$ so $$w^r(x,\cdot)$$ is uniformly bounded in $$H^3(Y)$$ from elliptic regularity (see McLean, 2000, Theorem 4.16). For $$h\in \mathbb{R}^3$$, we note that   −∇y⋅[b(x,y)∇y(wr(x+h,y)−wr(x,y))] =∇y⋅[(b(x+h,y)−b(x,y))er] +∇y⋅[(b(x+h,y)−b(x,y))∇ywr(x+h,y)]:=i1. As $$\int_Yw^r(x,y)\,{\rm d}y=0$$, we have   ‖wr(x+h,⋅)−wr(x,⋅)‖H1(Y) ≤c‖∇y(wr(x+h,⋅)−wr(x,⋅))‖L2(Y) ≤c‖(b(x+h,⋅)−b(x,⋅))er‖L2(Y) +c‖(b(x+h,⋅)−b(x,⋅))∇ywr(x+h,⋅)‖L2(Y), which converges to 0 when $$|h|\to 0$$. Fixing $$x\in\bar D$$, we then have from McLean (2000, Theorem 4.16) that   ‖wr(x+h,⋅)−wr(x,⋅)‖H3(Y)≤‖wr(x+h,⋅)−wr(x,⋅)‖H1(Y)+‖i1(x,⋅)‖H1(Y), (5.10) which converges to 0 when $$|h|\to 0$$. Fixing an index $$q=1,2,3$$, let $$h\in \mathbb{R}^3$$ be a vector whose components are all zero except the $$q$$th component. Let $$\eta(x,\cdot)\in H^1_\#(Y)/\mathbb{R}$$ be the solution of the problem   −∇y⋅[b(x,y)∇yη(x,y)]=∇y⋅[∂b∂xqer]+∇y⋅[∂b∂xq∇ywr(x,y)]. We have    −∇y⋅[b(x,y)∇y(wr(x+h,y)−wr(x,y)h−η(x,y))]  =∇y⋅[(b(x+h,y)−b(x,y)h−∂b∂xq(x,y))er]  +∇y⋅[(b(x+h,y)−b(x,y)h−∂b∂xq(x,y))∇ywr(x+h,y)]  +∇y⋅[∂b(x,y)∂xq(∇ywr(x+h,y)−∇ywr(x,y))]:=i2. From (5.10) and the regularity of $$b$$, $$\lim_{|h|\to 0}\|i_2(x,\cdot)\|_{H^1(Y)}=0$$. As $$\int_Yw^r(x,y)\,{\rm d}y=0$$ and $$\int_Y\eta(x,y)\,{\rm d}y=0$$, we have   lim|h|→0‖wr(x+h,⋅)−wr(x,⋅)h−η(x,⋅)‖H1(Y)=0. Therefore from McLean (2000, Theorem 4.16), we have   ‖wr(x+h,⋅)−wr(x,⋅)h−η(x,⋅)‖H3(Y)≤‖wr(x+h,⋅)−wr(x,⋅)h−η(x,⋅)‖H1(Y)+‖i2(x,⋅)‖H1(Y), which converges to 0 when $$|h|\to 0$$. Thus, $$w^r\in C^1(\bar D,H^3(Y))\subset C^1(\bar D,C^1(\bar Y))$$. □ For the regularity of the solution $$u_0$$ of the homogenized problem (4.7) we have the following result. Proposition 5.5 Assume that $$D$$ is a Lipschitz polygonal domain, and the coefficient $$a(\cdot,y)$$, as a function of $$x$$, is Lipschitz, uniformly with respect to $$y$$; then there is a constant $$0<s<1$$ so that $${\rm curl\,} u_0\in H^s(D)$$. Proof. When $$a(x,y)$$ is Lipschitz with respect to $$x$$, from (4.4), $$\|{\rm curl}_y N^r(x,\cdot)\|_{L^2(Y)}$$ is a Lipschitz function of $$x$$, so from (4.6) we have that $$a^0$$ is Lipschitz with respect to $$x$$. As $$a^0$$ is positive definite, $$(a^0)^{-1}$$ is Lipschitz. Let $$U=a^0{\rm curl\,} u_0$$. We have from (4.7) that $$U\in H({\rm curl\,},D)$$, $${\rm div}((a^0)^{-1}U)=0$$ and $$(a^0)^{-1}U\cdot \nu=0$$ on $$\partial D,$$ where $$\nu$$ is the outward normal vector on $$\partial D$$. The conclusion follows from Hiptmair (2002, Lemma 4.2). □ Remark 5.6 If $$a^0$$ is isotropic, we have from (4.7) that   curlcurlu0=−(a0)−1∇a0×curlu0−(a0)−1b0u0+(a0)−1f∈L2(D)3 so $$u_0\in H^1({\rm curl},D)$$. However, even if $$a$$ is isotropic, $$a^0$$ may not be isotropic. Remark 5.7 The homogenized equation for the multiscale case is determined as follows. We denote by $$a^n(x,\boldsymbol{y})=a(x,\boldsymbol{y})$$. Recursively, for $$i=1,\ldots,n-1$$, the $$i$$-th level homogenized coefficient is determined as follows. For $$r=1,2,3$$, let $$N_{i+1}^r\in L^2(D\times{\bf Y}_i,\tilde H({\rm curl}_{y_{i+1}},Y_{i+1}))$$ be the solution of the cell problem   ∫D∫Y1…∫Yi+1ai+1(x,yi,yi+1)(er+curlyi+1Ni+1r)⋅curlyi+1ψdyi+1dyidx=0 for all $$\psi\in L^2(D\times{\bf Y}_i,\tilde H({\rm curl}_{y_{i+1}},Y_{i+1}))$$. The $$i$$-th level homogenized coefficient $$a^{i}$$ is determined by   arsi(x,yi)=∫Yi+1ai+1(x,yi,yi+1)(es+curlyi+1Ni+1s)⋅(er+curlyi+1Ni+1r)dyi+1. Let $$b^n(x,\boldsymbol{y})=b(x,\boldsymbol{y})$$. Similarly, let $$w_{i+1}^{r}\in L^2(D\times{\bf Y}_i,H^1_\#(Y_{i+1})/\mathbb{R})$$ be the solution of the problem   ∫D∫Y1…∫Yi+1bi+1(x,yi,yi+1)(er+∇yi+1wi+1r)⋅∇yi+1ψdyi+1dyidx=0 for all $$\psi\in L^2(D\times{\bf Y}_i,H^1_\#(Y_{i+1})/\mathbb{R})$$. The $$i$$-th level homogenized coefficient $$b^i$$ is determined by   brsi(x,yi)=∫Yi+1bi+1(x,yi,yi+1)(es+∇yi+1wi+1s)⋅(er+∇yi+1wi+1r)dyi+1. We then have the equation    ∫D∫Yi[ai(x,yi)(curlu0+curly1u1+⋯+curlyiui)⋅(curlv0+curly1v1+⋯+curlyivi)dyidx  +bi(x,yi)(u0+∇y1u1+⋯+∇yiui)⋅(v0+∇y1v1+⋯+∇yivi)]dyidx=∫Df(x)⋅v0(x)dx. The coefficients $$a^0(x)$$ and $$b^0(x)$$ are the homogenized coefficients. We have   ui(x,yi) = [curlu0(x)r+curly1u1(x,y1)r+⋯+curlyi−1ui−1(x,yi−1)r]Nir(x,yi) =curlu0(x)r0(δr0r1+curly1N1r0(x,y1)r1)(δr1r2+curly2N2r1(x,y2)r2)⋯ (δri−2ri−1+curlyi−1Ni−1ri−2(x,yi−1)ri−1)Niri−1(x,yi). If $$a(x,\boldsymbol{y})\in C^1(\bar D,C^2(\bar Y_1,\ldots,C^2(\bar Y_n),\ldots))^{3\times 3}$$, by following the same procedure as above, we can show inductively that $${\rm curl}_{y_i}N_i^r(x,\boldsymbol{y}_i)\in C^1(\bar D,C^2(\bar Y_1,\ldots,C^2(\bar Y_{i-1},H^2(Y_i)),\ldots))$$ and $$a^i(x,\boldsymbol{y}_i)\in C^1(\bar D,C^2(\bar Y_1,\ldots,C^2(\bar Y_i),\ldots))$$. Thus, if $$u_0\in H^s({\rm curl\,},D)$$ for $$0<s\le 1$$, $$u_i\in \hat{\mathcal{H}}_i^s$$. Similarly, we can show that if $$b\in C^1(\bar D, C^2(\bar Y_1,\ldots,C^2(\bar Y_n)\ldots))$$ then $$w^{ir}\in C^1(\bar D,C^2(\bar Y_1,\ldots,C^2(\bar Y_{i-1},H^3(Y_i))\ldots))$$. As   ui=u0r0(x)(δr0r1+∂w1r0∂y1r1(x,y1))…(δri−2ri−1+∂wi−1ri−2∂y(i−1)ri−1(x,yi−1))wiri−1(x,yi), if $$u_0\in H^s(D)$$, $${\frak u_i}\in \hat{\frak H}_i^s$$. 6. Numerical results The detail spaces $$\mathcal{V}^l$$ and $$\mathcal{V}^l_\#$$, which are difficult to construct in numerical implementations, are defined via orthogonal projection in Section 3.2. We employ Riesz basis functions and define equivalent norms, which facilitate the construction of these spaces. We make the following assumption. Assumption 6.1 (i) For each multidimensional vector $$j \in \mathbb{N}_0^d$$, there exists a set of indices $$I^j \subset \mathbb{N}^d_0$$ and a set of basis functions $$\phi^{jk}\in L^2(D)$$ for $$k\in I^j$$, such that $$V^l = \text{span}\left\{\phi^{jk} : |\,j|_{\infty}\le l\right\}$$. There are constants $$c_2>c_1>0$$ such that if $$\phi = \sum_{|\,j|_{\infty}\leq l,k\in I^j}\phi^{jk}c_{jk}\in V^l$$, then the following norm equivalences hold:   c1∑|j|∞≤lk∈Ij|cjk|2≤‖ϕ‖L2(D)2≤c2∑|j|∞≤lk∈Ij|cjk|2, where $$c_1$$ and $$c_2$$ are independent of $$\phi$$ and $$l$$. (ii) For the space $$L^2(Y)$$, for each $$j \in \mathbb{N}_0^d$$, there exists a set of indices $$I^j_0 \subset \mathbb{N}_0^d$$ and a set of basis functions $$\phi^{jk}_0\in L^2(Y)$$, $$k\in I^j_0$$, such that $$V^l_\# = \text{span}\{\phi^{jk}_0 : |\,j|_{\infty}\le l\}$$. There are constants $$c_4>c_3>0$$ such that if $$\phi = \sum_{|\,j|_{\infty}\leq l,k\in I^{\,j}_{\,0}}\phi^{jk}_0c_{jk}\in V^l$$ then   c3∑|j|∞≤lk∈I0j|cjk|2≤‖ϕ‖L2(Y)2≤c4∑|j|∞≤lk∈I0j|cjk|2, where $$c_3$$ and $$c_4$$ are independent of $$\phi$$ and $$l$$. Because of the norm equivalence, we can use $$\mathcal{V}^l=\text{span}\{\phi^{jk} : |\,j|_{\infty}= l\}$$ and $$\mathcal{V}^l_\#=\text{span}\{\phi^{jk}_0 : |\,j|_{\infty}= l\}$$ to construct the sparse tensor product FE spaces. Example 6.2 (i) We can construct a hierarchical basis for $$L^2(0,1)$$ as follows. We first take three piecewise linear functions as the basis for level $$j=0$$: $$\psi^{01}$$ obtains values $$(1,0)$$ at $$(0,1/2)$$ and is 0 in $$(1/2,1)$$, $$\psi^{02}$$ is piecewise linear and obtains values $$(0, 1, 0)$$ at $$(0, 1/2, 1)$$ and $$\psi^{03}$$ obtains values $$(0,1)$$ at $$(1/2, 1)$$ and is 0 in $$(0,1/2)$$. The basis functions for other levels are constructed from the wavelet function $$\psi$$ that takes values $$(0,-1,2,-1,0)$$ at $$(0,1/2,1,3/2,2)$$, the left boundary function $$\psi^{\rm left}$$ taking values $$(-2,2,-1,0)$$ at $$(0,1/2,1,3/2)$$ and the right boundary function $$\psi^{\rm right}$$ taking values $$(0, -1,2,-2)$$ at $$(1/2,1,3/2,2)$$. For levels $$j\geq 1$$, $$I^j=\{1,2,\ldots,2^j\}$$. The wavelet basis functions are defined as $$\psi^{j1}(x) = 2^{j/2}\psi^{\rm left}(2^j x)$$, $$\psi^{jk}(x)=2^{j/2}\psi(2^j x - k + 3/2)$$ for $$k = 2, \ldots, 2^j-1$$ and $$\psi^{j2^j} = 2^{j/2}\psi^{\rm right}(2^j x - 2^j+2)$$. This base satisfies Assumption 6.1 (i). (ii) For $$Y = (0,1)$$, we can construct a hierarchy of periodic basis functions for $$L^2(Y)$$ that satisfies Assumption 6.1 (ii) from those in (i). For level 0, we exclude $$\psi^{01}$$, $$\psi^{03}$$ and include the periodic piecewise linear function that takes values $$(1,0,1)$$ at $$(0,1/2,1),$$ respectively. At other levels, the functions $$\psi^{\rm left}$$ and $$\psi^{\rm right}$$ are replaced by the piecewise linear functions that take values $$(0,2, -1, 0)$$ at $$(0,1/2,1,3/2)$$ and values $$(0, -1,2,0)$$ at $$(1/2,1, 3/2 ,2),$$ respectively. When $$D=(0,1)^d$$, the basis functions can be constructed by taking the tensor products of the basis functions in $$(0,1)$$. They satisfy Assumption 6.1 after appropriate scaling (see Griebel & Oswald, 1995). Remark 6.3 When the norm equivalence for the basis functions in $$L^2(D)$$ and in $$L^2(Y)$$ does not hold, in many cases, we can still prove a rate of convergence similar to those in Lemmas 3.5 and 3.6 for the sparse tensor product FE approximations. For example, with the division of the domain $$D$$ into sets of triangles $$\mathcal{T}^l$$, the set of continuous piecewise linear functions with value 1 at one vertex and 0 at all the others forms a basis of $$V^l$$. Let $$S^l$$ be the set of vertices of the set of simplices $$\mathcal{T}^l$$. We can define $$\mathcal{V}^l$$ as the linear span of functions that are 1 at a vertex in $$S^l\setminus S^{l-1}$$ and 0 at all the other vertices. We can then construct the sparse tensor product FE approximations with these spaces but the norm equivalence does not hold. A rate of convergence for sparse tensor product FEs similar to those in Lemmas 3.5 and 3.6 can be deduced (see e.g, Hoang, 2008). In the first example, we consider a two-scale Maxwell-type equation in the two-dimensional domain $$D=(0,1)^2$$. The coefficients   a(x,y)=(1+x1)(1+x2)(1+cos2⁡2πy1)(1+cos2⁡2πy2) and   b(x,y)=1(1+x1)(1+x2)(1+cos2⁡2πy1)(1+cos2⁡2πy2). We can compute the homogenized coefficients exactly. In this case,   a0=4(1+x1)(1+x2)9  and  b0=23(1+x1)(1+x2). We choose   f=(49(1+x1)(1+2x2−x1)+23(1+x1)(1+x2)x1x2(1−x2)49(1+x2)(1+2x1−x2)+23(1+x1)(1+x2)x1x2(1−x1)) so that the solution to the homogenized equation is   u0=(x1x2(1−x2)x1x2(1−x1)). In Fig. 1, we plot the energy error versus the mesh size for the sparse tensor product FE approximations of the two-scale homogenized Maxwell-type problem. The figure agrees with the error estimate in Proposition 3.8. Fig. 1. View largeDownload slide The sparse tensor energy error $$B(\boldsymbol{u}-\widehat {\bf{u}}^L,\boldsymbol{u}-\widehat {\bf{u}}^L).$$ Fig. 1. View largeDownload slide The sparse tensor energy error $$B(\boldsymbol{u}-\widehat {\bf{u}}^L,\boldsymbol{u}-\widehat {\bf{u}}^L).$$ In the second example, we consider the case where $$b$$ is the identity matrix, i.e., it does not depend on $$y$$. In this case, from (2.10) we note that the function $${\frak u_1}=0$$. We choose   a(x,y)=(1+x1)(1+x2)(1+cos2⁡2πy1)(1+cos2⁡2πy2) and   f=(4(2π(1+x1)(1+x2)sin⁡2πx2+(1+x1)(cos⁡2πx1−cos⁡2πx2))9+12πsin⁡2πx24(2π(1+x1)(1+x2)sin⁡2πx1−(1+x2)(cos⁡2πx1−cos⁡2πx2))9+12πsin⁡2πx1) so that the solution to the homogenized problem is   u0=(12πsin⁡2πx212πsin⁡2πx1). Figure 2 plots the energy error versus the mesh size for the sparse tensor product FE approximations for the two-scale homogenized Maxwell-type problem. The plot confirms the analysis. Fig. 2. View largeDownload slide The sparse tensor energy error $$B(\boldsymbol{u}-\widehat {\bf{u}}^L,\boldsymbol{u}-\widehat {\bf{u}}^L).$$ Fig. 2. View largeDownload slide The sparse tensor energy error $$B(\boldsymbol{u}-\widehat {\bf{u}}^L,\boldsymbol{u}-\widehat {\bf{u}}^L).$$ Acknowledgements The authors gratefully acknowledge a postgraduate scholarship of Nanyang Technological University, the AcRF Tier 1 grant RG69/10, the Singapore A*Star SERC grant 122-PSF-0007 and the AcRF Tier 2 grant MOE 2013-T2-1-095 ARC 44/13. Footnotes 1 The notations $$Y_1,\ldots,Y_n$$, which denote the same unit cube $$Y$$, are introduced for convenience only, especially in the case where the Cartesian product of several of them is used, to avoid the necessity of indicating how many times the unit cube appears in the product. The functions $$a$$ and $$b$$ depend on the macroscopic scale only and are periodic with respect to $$y_i$$ with the period being the unit cube $$Y$$. 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( 2010) Multiscale computations for 3D time-dependent Maxwell’s equations in composite materials. SIAM J. Sci. Comput. , 32, 2560– 2583. Google Scholar CrossRef Search ADS   Appendix A. We present the proof of Theorem 4.2 in this appendix. We consider a set of $$M$$ open cubes $$Q_i$$ ($$i=1,\ldots,M$$) of size $$\varepsilon^t$$ for $$t>0$$ to be chosen later such that $$D\subset\bigcup_{i=1}^MQ_i$$ and $$Q_i\bigcap D\ne\emptyset$$. Each cube $$Q_i$$ intersects with only a finite number, which does not depend on $$\varepsilon$$, of other cubes. We consider a partition of unity that consists of $$M$$ functions $$\rho_i$$ such that $$\rho_i$$ has support in $$Q_i$$, $$\sum_{i=1}^M\rho_i(x)=1$$ for all $$x\in D$$ and $$|\nabla\rho_i(x)|\le c\varepsilon^{-t}$$ for all $$x$$ (indeed such a set of cubes $$Q_i$$ and a partition of unity can be constructed from a fixed set of cubes of size $${\mathcal O}(1)$$ by rescaling). For $$r=1,2,3$$ and $$i=1,\ldots,M$$, we define   Uir=1|Qi|∫Qicurlu0(x)rdx and   Vir=1|Qi|∫Qiu0(x)rdx (as $$u_0\in H^s(D)^3$$ and $${\rm curl\,} u_0\in H^s(D)^3$$, for the Lipschitz domain $$D$$, we can extend each of them, separately, continuously outside $$D$$ and understand $$u_0$$ and $${\rm curl\,} u_0$$ as these extensions; see Wloka, 1987, Theorem 5.6). Let $$U_i$$ and $$V_i$$ denote the vectors $$(U_i^1, U_i^2,U_i^3)$$ and $$(V_i^1,V_i^2,V_i^3),$$ respectively. Let $$B$$ be the unit cube in $$\mathbb{R}^3$$. From the Poincaré inequality, we have   ∫B|ϕ−∫Bϕ(x)dx|2dx≤c∫B|∇ϕ(x)|2dx  ∀ϕ∈H1(B). By translation and scaling, we deduce that   ∫Qi|ϕ−1|Qi|∫Qiϕ(x)dx|2dx≤cε2t∫Qi|∇ϕ(x)|2dx  ∀ϕ∈H1(Qi), i.e.,   ‖ϕ−1|Qi|∫Qiϕ(x)dx‖L2(Qi)≤cεt‖ϕ‖H1(Qi). Together with   ‖ϕ−1|Qi|∫Qiϕ(x)dx‖L2(Qi)≤c‖ϕ‖L2(Qi), we deduce from interpolation that   ‖ϕ−1|Qi|∫Qiϕ(x)dx‖L2(Qi)≤cεts‖ϕ‖Hs(Qi)  ∀ϕ∈Hs(Qi). Thus,   ∫Qi|curlu0(x)r−Uir|2dx≤cε2ts‖(curlu0)r‖Hs(Qi)2. (A.1) Let   u1ε(x)=u0(x)+εNr(x,xε)Ujrρj(x)+ε∇[wr(x,xε)Vjrρj(x)]. We have   curl(aε(x)curlu1ε(x))+bε(x)u1ε(x) =curla(x,xε)[curlu0(x)+εcurlxNr(x,xε)Ujrρj(x)+curlyNr(x,xε)Ujrρj+ε(Ujr∇ρj)×Nr(x,xε)] +b(x,xε)[u0(x)+εNr(x,xε)Ujrρj(x)+ε∇xwr(x,xε)Vjrρj(x)+∇ywr(x,xε)Vjrρj(x)  + εwr(x,xε)Vjr∇ρj(x)] =curl(a0(x)curlu0(x))+b0(x)u0(x)+curl[Gr(x,xε)Ujrρj(x)]+gr(x,xε)Vjrρj(x)+εcurlI(x) +εJ(x)+curl[(aε(x)−a0(x))(curlu0(x)−Ujρj(x))]+(bε(x)−b0(x))(u0(x)−Vjρj(x)), where the vector functions $$G_r(x,y)$$ and $$g_r(x,y)$$ are defined by   (Gr)i(x,y) =air(x,y)+aij(x,y)curlyNr(x,y)j−air0(x), (A.2)  (gr)i(x,y) =bir(x,y)+bij(x,y)∂wr∂yj(x,y)−bir0(x) (A.3) and   I(x) =a(x,xε)[curlxNr(x,xε)Ujrρj(x)+(Ujr∇ρj(x))×Nr(x,xε)],J(x) =b(x,xε)[Nr(x,xε)Ujrρj(x)+∇xwr(x,xε)Vjrρj(x)+wr(x,xε)Vjr∇ρj(x)]. Therefore, for $$\phi\in W$$,    ⟨curl(aεcurlu1ε)+bεu1ε−curl(a0curlu0)−b0u0,ϕ⟩  =∫DUjrρj(x)Gr(x,xε)⋅curlϕdx+∫DVjrρj(x)gr(x,xε)⋅ϕ(x)dx  +ε∫DI(x)⋅curlϕ(x)dx+ε∫DJ(x)⋅ϕ(x)dx+∫D(aε−a0)(curlu0(x)−Ujρj)⋅curlϕ(x)dx +∫D(bε−b0)(u0−Vjρj)⋅ϕdx (here $$\langle\cdot\rangle$$ denotes the duality pairing between $$W'$$ and $$W$$). From (4.4), we have that $${\rm curl}_yG_r(x,y)=0$$. Further, from (4.6) $$\int_YG_r(x,y)\,{\rm d}y=0$$. Therefore, there is a function $$\tilde G_r(x,y)$$ such that $$G_r(x,y)=\nabla_y \tilde G_r(x,y)$$. From (4.1), we have $${\rm div}_yg_r(x,y)=0$$ and from (4.6) $$\int_Yg_r(x,y)\,{\rm d}y=0$$. Hence, there is a function $$\tilde g_r$$ such that $$g_r(x,y)={\rm curl}_y\tilde g_r(x,y)$$. As $$\nabla_y\tilde G_r(x,\cdot)=G_r(x,\cdot)\in H^1(Y)^3$$ so $$\Delta_y\tilde G_r(x,\cdot)\in L^2(Y)$$. Thus, $$\tilde G_r(x,\cdot)\in H^2(Y),$$ which implies $$\tilde G_r(x,\cdot)\in C(\bar Y)$$. As $$G_r(x,\cdot)\in C^1(\bar D,H^1_\#(Y)^3)$$, we deduce that $$\tilde G_r(x,y)\in C^1(\bar D, H^2(Y))\subset C^1(\bar D,C(\bar Y))$$. The construction of $$\tilde g_r$$ in Jikov et al. (1994) implies that $$\tilde g_r\in C^1(\bar D, C(\bar Y))$$ (see Hoang & Schwab, 2013). We have    ∫DUjrρjGr(x,xε)⋅curlϕdx=∫DUjrρj(x)[ε∇G~r(x,xε)−ε∇xG~r(x,xε)]⋅curlϕdx =−ε∫DG~r(x,xε)div[(Ujrρj)curlϕ]dx−ε∫DUjrρj∇xG~r(x,xε)⋅curlϕdx. We note that   |∫DUjrρj∇xG~r(x,xε)⋅curlϕdx|≤c‖(Ujrρj)‖L2(D)‖curlϕ‖L2(D)3. From   ‖Ujrρj‖L2(D)2=∫D(Ujr)2ρj(x)2dx+∑i≠j∫DUirUjrρi(x)ρj(x)dx, and the fact that the support of each function $$\rho_i$$ intersects only with the support of a finite number (which does not depend on $$\varepsilon$$) of other functions $$\rho_j$$ in the partition of unity, we deduce   ‖Ujrρj‖L2(D)2 ≤c∑j=1M(Ujr)2|Qj| =c∑j=1M1|Qj|(∫Qjcurlu0(x)rdx)2≤c∑j=1M∫Qjcurlu0(x)r2dx≤c∫Dcurlu0(x)r2dx. Thus,   |ε∫DUjrρj∇xG~r(x,xε)⋅curlϕdx|≤cε‖curlϕ‖L2(D)3. We have further that   ε∫DG~r(x,xε)div[(Ujrρj)curlϕ]dx =ε∫DG~r(x,xε)[(Ujr∇ρj(x))]⋅curlϕdx ≤cε‖Ujr∇ρj‖L2(D)3‖curlϕ‖L2(D)3. As the support of each function $$\rho_i$$ intersects with the support of a finite number of other functions $$\rho_j$$ and $$\|\nabla\rho_j\|_{L^\infty(D)}\le c\varepsilon^{-t}$$, we have   ‖Ujr∇ρj‖L2(D)32≤c∑j=1M(Ujr)2|Qj|‖∇ρj‖L∞(D)2≤cε−2t∑j=1M(Ujr)2|Qj|≤cε−2t, so   ε∫DG~r(x,xε)div[(Ujrρj)curlϕ]dx≤cε‖Ujr∇ρj‖L2(D)3‖curlϕ‖L2(D)3≤cε1−t‖curlϕ‖L2(D)3. We therefore deduce that   |∫DUjrρjGr(x,xε)⋅curlϕdx|≤cε1−t‖curlϕ‖L2(D)3. We have   ∫DVjrρjgr(x,xε)⋅ϕ(x)dx=∫DVjrρj[εcurlg~r(x,xε)−εcurlxg~r(x,xε)]⋅ϕdx. Arguing similarly to above, we have   |ε∫DVjrρjcurlxg~r(x,xε)⋅ϕdx|≤cε‖Vjrρj‖L2(D)3‖ϕ‖L2(D)3≤cε‖ϕ‖L2(D)3 and   |ε∫DVjrρjcurlg~r(x,xε)⋅ϕdx| = |ε∫Dg~r(x,xε)⋅curl[(Vjrρj)ϕ]dx| ≤ |ε∫Dg~r(x,xε)⋅[(Vjrρj)curlϕ+ϕ×(Vjr∇ρj)]dx ≤c(ε‖curlϕ‖L2(D)3+cε1−t‖ϕ‖L2(D)3)(∑j=1M(Vjr)2|Qj|)1/2 ≤c(ε‖curlϕ‖L2(D)3+cε1−t‖ϕ‖L2(D)3). We note that   ‖I‖L2(D)3≤csupr[‖Ujrρj‖L2(D)+‖Ujr∇ρj‖L2(D)]≤cε−t and   ‖J‖L2(D)3≤csupr[‖Ujrρj‖L2(D)+c‖Vjrρj‖L2(D)+c‖Vjr∇ρj‖L2(D)]≤cε−t. We have further that   ⟨curl((aε−a0)(curlu0−Ujρj)),ϕ⟩≤c‖curlu0−(Ujρj))‖L2(D)3‖curlϕ‖L2(D)3. From   ∫D|(curlu0)r−(Ujrρj)|2dx=∫D|∑j=1M((curlu0)r−Ujr)ρj|2dx, using the support property of $$\rho_j$$, we have from (A.1),   ∫D|(curlu0)r−(Ujrρj)|2dx ≤c∑j=1M∫Qj|(curlu0)r−Ujr|2dx≤cε2st∑j=1M‖(curlu0)r‖Hs(Qj)2 =cε2st∑j=1M[∫Qj(curlu0)r2dx+∫Qj×Qj(curlu0(x)r−curlu0(x′)r)2|x−x′|3+2sdxdx′] ≤cε2st[‖(curlu0)r‖L2(D)2+∫D×D(curlu0(x)r−curlu0(x′)r)2|x−x′|3+2sdxdx′] =cε2st‖(curlu0)r‖Hs(D)2. (A.4) Thus,   ⟨curl((aε−a0)(curlu0−Ujρj)),ϕ⟩≤cεst‖curlϕ‖L2(D)3. Similarly, we have   |∫D(bε−b0)(u0−∑j=1MVjρj)⋅ϕdx|≤c‖∑j=1M(u0−Vj)ρj‖L2(D)3‖ϕ‖L2(D)3≤cεst‖ϕ‖L2(D)3. Therefore,   |⟨curl(aεcurlu1ε)+bεu1ε−curl(a0curlu0)−b0u0,ϕ⟩|≤c(ε1−t+εst)‖ϕ‖V i.e.,   ‖curl(aεcurlu1ε)+bεu1ε−curl(a0curlu0)−b0u0‖W′≤c(ε1−t+εst). Thus,   ‖curl(aεcurlu1ε)+bεu1ε−curl(aεcurluε)−bεuε‖W′≤c(ε1−t+εst). (A.5) Let $$\tau^\varepsilon(x)$$ be a function in $$\mathcal{D}(D)$$ such that $$\tau^\varepsilon(x)=1$$ outside an $$\varepsilon$$ neighbourhood of $$\partial D$$ and $${\rm sup}_{x\in D}\varepsilon|\nabla\tau^\varepsilon(x)|<c,$$ where $$c$$ is independent of $$\varepsilon$$. We consider the function   w1ε(x)=u0(x)+ετε(x)Ujrρj(x)Nr(x,xε)+ε∇[Vjrρjτε(x)wr(x,xε)]. We then have   u1ε−w1ε=ε(1−τε(x))Ujrρj(x)Nr(x,xε)+ε∇[(1−τε(x))Vjrρjwr(x,xε)] and   curl(u1ε−w1ε) =εcurlxNr(x,xε)Ujrρj(x)(1−τε(x))+curlyNr(x,xε)Ujrρj(x)(1−τε(x)) −εUjrρj(x)∇τε(x)×Nr(x,xε)+ε(1−τε(x))Ujr∇ρj(x)×Nr(x,xε). As shown above, $$\|U_j^r\rho_j\|_{L^2(D)}$$ is uniformly bounded, so   ‖εcurlxNr(x,xε)(Ujrρj)(1−τε(x))‖L2(D)3≤cε. Let $$\tilde D^\varepsilon$$ be the $$3\varepsilon^{t}$$ neighbourhood of $$\partial D$$. We note that $${\rm curl\,} u_0$$ is extended continuously into a function in $$H^s(\mathbb{R}^3)$$ outside $$D$$. As shown in Hoang & Schwab (2013), for $$\phi\in H^1(\tilde D^\varepsilon)$$,   ‖ϕ‖L2(D~ε)≤cεt/2‖ϕ‖H1(D~ε). From this and   ‖ϕ‖L2(D~ε)≤‖ϕ‖L2(D~ε), using interpolation, we get   ‖ϕ‖L2(D~ε)≤cεst/2‖ϕ‖Hs(D~ε)≤cεst/2‖ϕ‖Hs(D) for all $$\phi\in H^s(D)$$ extended continuously outside $$D$$. We then have   ‖Ujrρj‖L2(Dε)2 ≤c∑j=1M∫Qj⋂Dε(Ujr)2ρj2dx ≤c∑j=1M|Qj⋂Dε|1|Qj|2(∫Qj(curlu0)rdx)2 ≤c∑Qj⋂Dε≠∅|Qj⋂Dε||Qj|∫Qj(curlu0)r2dx. As $$D^\varepsilon$$ is the $$\varepsilon$$ neighbourhood of $$\partial D$$, $$\partial D$$ is Lipschitz and $$Q_j$$ has size $$\varepsilon^t$$, $$|Q_j\bigcap D^\varepsilon|\le c\varepsilon^{1+(d-1)t}$$ so $$|Q_j\bigcap D^\varepsilon|/|Q_j|\le c\varepsilon^{1-t}$$. When $$Q_j\bigcap D^\varepsilon\ne\emptyset$$, $$Q_j\subset\tilde D^\varepsilon$$. Thus,   ‖Ujrρj‖L2(Dε)2≤cε1−t‖(curlu0)r‖L2(D~ε)2≤cε1−t+st‖curlu0‖Hs(D)32. Therefore,   ‖curlyNr(x,xε)(Ujrρj)(1−τε(x))‖L2(D)3≤cε(1−t+st)/2 and   ‖ε(Ujrρj)∇τε(x)×Nr(x,xε)‖L2(D)3≤cε(1−t+st)/2. Similarly, we have   ‖Ujr∇ρj‖L2(Dε)32 ≤cε−2t∑Qj⋂Dε≠∅|Qj⋂Dε||Qj|∫Qj(curlu0)r2dx ≤cε−2t+1−t‖curlu0‖L2(D~ε)32≤cε1−3t+st‖curlu0‖Hs(D)32. Thus,   ‖ε(1−τε(x))(Ujr∇ρj)×Nr(x,xε)‖L2(D)≤cε(1−t)+(1−t+st)/2. Therefore,   ‖curl(u1ε−w1ε)‖L2(D)3≤c(ε(1−t+st)/2+ε(1−t)+(1−t+st)/2). We further have   ε∇[(1−τε(x))wr(x,xε)(Vjrρj)] = −ε∇τε(x)wr(x,xε)(Vjrρj)+ε(1−τε(x))∇xwr(x,xε)(Vjrρj) +(1−τε(x))∇ywr(x,xε)(Vjrρj)+ε(1−τε(x))wr(x,xε)(Vjr∇ρj). Arguing as above, we deduce that   ‖Vjrρj‖L2(Dε)≤cε(1−t+st)/2,  ‖Vjr∇ρj‖L2(Dε)≤cε(1−t+st)/2−t. Therefore,   ‖ε∇[(1−τε(x))wr(x,xε)(Vjrρj)]‖L2(D)3≤c(ε(1−t+st)/2+ε1−t+(1−t+st)/2). Thus,   ‖u1ε−w1ε‖L2(D)3≤c(ε(1−t+st)/2+ε(1−t)+(1−t+st)/2). Choosing $$t=1/(s+1)$$ we have   ‖curl(aεcurl(u1ε−w1ε))+bε(u1ε−w1ε)‖W′≤cεs/(s+1). This together with (A.5) gives   ‖curl(aεcurl(uε−w1ε))+bε(uε−w1ε)‖W′≤cεs/(s+1). Thus,   ‖uε−w1ε‖W≤cεs/(s+1), which implies   ‖uε−u1ε‖W≤cεs/(s+1). (A.6) We note that   curlu1ε=curlu0(x)+curlyNr(x,xε)(Ujrρj)+εcurlxNr(x,xε)(Ujrρj)+ε(Ujr∇ρj)×Nr(x,xε). From   ‖εcurlxNr(x,xε)(Ujrρj)‖L2(D)3≤cε  and  ‖ε(Ujr∇ρj)×Nr(x,xε)‖L2(D)3≤cεε−t=cεs/(1+s), we deduce that   ‖curlu1ε−curlu0−curlyNr(x,xε)(Ujrρj)‖L2(D)3≤cεs/(s+1). From (A.4),   ‖curlu0−(Ujrρj)‖L2(D)3≤cεts=cεs/(s+1), we get   ‖curlu1ε−[curlu0+curlyNr(x,xε)(curlu0)r‖L2(D)3≤cεs/(s+1). This together with (A.6) implies   ‖curluε−[curlu0+curlyNr(x,xε)(curlu0)r‖L2(D)3≤cεs/(s+1). □ Appendix B. We prove Lemma 4.3 in this appendix. We adapt the proof of Hoang & Schwab (2013, Lemma 5.5). As   u1(x,y)=∑r=13curlu0(x)rNr(x,y), it is sufficient to show that for each $$r=1,2,3$$,   ∫D|curlu0(x)rcurlyNr(x,xε)−∫Ycurlu0(ε[xε]+εt)rcurlyNr(ε[xε]+εt,xε)dt|2dx≤cε2s. The expression on the left-hand side is bounded by    ∫D∫Y|curlu0(x)rcurlyNr(x,xε)−curlu0(ε[xε]+εt)rcurlyNr(ε[xε]+εt,xε)|2dtdx ≤2∫D∫Y|(curlu0(x)r−curlu0(ε[xε]+εt)r)curlyNr(ε[xε]+εt,xε)|2dtdx +2∫D∫Y|curlu0(x)r|2|curlyNr(x,xε)−curlyNr(ε[xε]+εt,xε)|2dtdx. As $${\rm curl}_yN^r\in C^1(\bar D,C(\bar Y))^3$$, there exists a constant $$c$$ such that   supx∈Dsupt∈Y|curlyNr(x,xε)−curlyNr(ε[xε]+εt,xε)|≤cε. From this we have   ∫D|curlu0(x)rcurlyNr(x,xε)−Uε(curlu0(⋅)rcurlyNr(⋅,⋅))(x)|2dx≤c∫D∫Y|curlu0(x)r−curlu0(ε[xε]+εt)r|2dtdx+cε2. We now show that for $${\rm curl\,} u_0\in H^s(D)$$,   ∫D∫Y|curlu0(x)r−curlu0(ε[xε]+εt)r|2dtdx≤cε2s. (B.1) Letting $$\phi(x)$$ be a smooth function, we have    ∫D∫Y|ϕ(x)−ϕ(ε[xε]+εt)|2dtdx ≤∑i=1d∫D∫Y|ϕ(ε[x1ε]+εt1,…,ε[xi−1ε]+εti−1,xi,…,xd) −ϕ(ε[x1ε]+εt1,…,ε[xiε]+εti,xi+1,…,xd)|2dtdx ≤∑i=1d∫D∫Y|ε∫ti{xi/ε}∂ϕ∂xi(ε[x1ε]+εt1,…,ε[xiε]+εζi,xi+1,…,xd)dζi|2dtdx ≤ε2∑i=1d∫D∫Y∫01|∂ϕ∂xi(ε[x1ε]+εt1,…,ε[xiε]+εζi,xi+1,…,xd)|2dζidtdx ≤ε2∑i=1d∫D|∂ϕ∂xi|2dx, which follows from (4.13); here we freeze the variables $$x_{i+1},\ldots,x_d$$. Let $$\psi\in H^1(D)$$. We consider a sequence $$\{\phi_n\}_n\subset C^\infty(\bar D),$$ which converges to $$\psi$$ in $$H^1(D)$$. As $$n\rightarrow \infty$$,   ∫D∫Y(ϕn(ε[xε]+εt)−ψ(ε[xε]+εt))2dtdx = ∫DUε((ϕn−ψ)2)(x)dx ≤ ∫D(ϕn(x)−ψ(x))2dx→0. Therefore,   ∫D∫Y(ψ(x)−ψ(ε[xε]+εt))2dtdx ≤3∫D(ψ−ϕn)2dx+3∫D∫Y(ϕn−ϕn(ε[xε]+εt))2dtdx +3∫D∫Y(ϕn(ε[xε]+εt)−ψ(ε[xε]+εt))2dtdx ≤6∫D(ψ−ϕn)2dx+3ε2∑i=1d∫D|∂ϕn∂xi|2dx. Letting $$n\to\infty$$, we have   ∫D∫Y(ψ(x)−ψ(ε[xε]+εt))2dtdx≤3ε2∑i=1d∫D|∂ψ∂xi|2dx. Let $$T$$ be the linear map from $$L^2(D)$$ to $$L^2(D\times Y)$$ so that   T(ϕ)(x,y)=ϕ(x)−ϕ(ε[xε]+εt). We thus have   ‖T‖H1(D)→L2(D×Y)≤cε. On the other hand,   ‖T‖L2(D)→L2(D×Y)≤c. From interpolation theory, we deduce that   ‖T‖Hs(D)→L2(D×Y)≤cεs. We then get (B.1). The conclusion follows. □ Notes added after the proof stage: After the article is accepted, we learnt about the related recent article: P. Henning, M. Ohlberger and B. Verfürth (2016), A new heterogeneous multiscale method for time-harmonic Maxwell’s equations, SIAM J. Numer. Anal., 54, 3493–3522. This article considers a locally periodic two-scale time harmonic Maxwell equation, but the variational form is still assumed to be strictly coercive, uniformly with respect to the microscopic scale, similar to the equation considered in our present article. These authors formulate the two-scale homogenized equation in a slightly different manner. The Heterogeneous Multiscale Method (HMM) is used to solve the two-scale problem, and is shown to be equivalent to solving the two-scale homogenized equation by using the full tensor finite element spaces. © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IMA Journal of Numerical Analysis Oxford University Press

High-dimensional finite elements for multiscale Maxwell-type equations

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Oxford University Press
Copyright
© The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
ISSN
0272-4979
eISSN
1464-3642
D.O.I.
10.1093/imanum/drx001
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Abstract

Abstract We consider multiscale Maxwell-type equations in a domain $$D\subset\mathbb{R}^d$$ ($$d=2,3$$), which depend on $$n$$ microscopic scales. Using multiscale convergence, we derive the multiscale homogenized problem, which is posed in $$\mathbb{R}^{(n+1)d}$$. Solving it, we get all the necessary macroscopic and microscopic information. Sparse tensor product finite elements (FEs) are employed, using edge FEs. The method achieves a required level of accuracy with essentially an optimal number of degrees of freedom, which, apart from a multiplying logarithmic term, is equal to that for solving a problem in $$\mathbb{R}^d$$. Numerical correctors are constructed from the FE solutions. In the two-scale case, an explicit homogenization error is deduced. To get this error, the standard procedure in the homogenization literature requires the solution $$u_0$$ of the homogenized problem to belong to $$H^1({\rm curl\,},D)$$. However, in polygonal domains, $$u_0$$ belongs only to a weaker regularity space $$H^s({\rm curl\,},D)$$ for $$0<s<1$$. We derive a homogenization error estimate for this case. Though we prove the result for two-scale Maxwell-type equations, the approach works verbatim for elliptic and elasticity problems when the solution to the homogenized equation belongs to $$H^{1+s}(D)$$ (standard procedure requires $$H^2(D)$$ regularity). This homogenization error estimate is new in the literature. Thus, for two-scale problems, an explicit error for the numerical corrector is obtained; it is of the order of the sum of the homogenization error and the FE error. For the case of more than two scales, we construct a numerical corrector, albeit without a rate of convergence, as such a homogenization error is not available. Numerical experiments confirm the theoretical results. 1. Introduction We consider Maxwell-type equations that depend on $$n$$ separable microscopic scales in a domain $$D\in \mathbb{R}^d,$$ where $$d=2,3$$. The coefficients are assumed to be locally periodic with respect to each microscopic scale. We use the multiscale convergence of a bounded sequence in $$H({\rm curl\,},D)$$ to derive the multiscale homogenized equation, which contains all the necessary information. Solving it, we get the solution of the homogenized equation that describes the multiscale solution macroscopically and the scale interacting terms (the corrector terms) that encode the multiscale information. However, this equation is posed in high-dimensional product domains. It depends on $$n+1$$ variables in $$\mathbb{R}^d$$, one for each scale that the original multiscale problem depends on. The full tensor product finite element (FE) method requires a large number of degrees of freedom, and thus is prohibitively expensive. We develop the sparse tensor FE product approach, using edge FEs, for this multiscale homogenized Maxwell-type equation. The approach achieves accuracy essentially equal to that obtained by the full tensor product FEs but requires an essentially optimal level of complexity that is essentially equal to that for solving a problem in $$\mathbb{R}^d$$ only. Analytic homogenization for two-scale Maxwell-type equations is well developed. We mention the standard references Bensoussan et al. (1978), Sanchez-Palencia (1980) and Jikov et al. (1994). However, there has been little effort on numerical analysis of multiscale Maxwell-type equations. As for other multiscale problems, a direct numerical treatment needs a fine mesh which is at most of the order of the smallest scale, leading to a prohibitive level of complexity. The multiscale FE method (Hou & Wu, 1997; Efendiev & Hou, 2009) and the heterogeneous multiscale method (E & Engquist, 2003; Abdulle et al., 2012) are designed to overcome this difficulty but their applications to multiscale Maxwell-type equations have not been adequately studied. Solving cell problems to establish the homogenized equation and using the cell problems’ solutions to compute the correctors for two-scale Maxwell-type equations are performed in Zhang et al. (2010). However, as for other multiscale problems, this approach is rather expensive, especially when the coefficients are only locally periodic, as for each macroscopic point, several cell problems need to be solved. We contribute in this article a feasible general numerical method for locally periodic multiscale Maxwell-type problems. We employ the sparse tensor product FE approach developed by Hoang & Schwab (2004/05) for multiscale elliptic equations (see also Hoang, 2008; Harbrecht & Schwab, 2011; Xia & Hoang, 2014, 2015a,b). It achieves the required level of accuracy with an essentially optimal number of degrees of freedom. We note that sparse tensor edge FEs are considered in Hiptmair et al. (2013) in the context of computing the moments of the solutions to stochastic Maxwell-type problems. However, our setting is quite different, and does not require constructing the detail spaces for edge FEs. We only need the detail spaces for the nodal FEs that approximate functions in the Lebesgue spaces $$L^2$$. We then construct a numerical corrector for the solution of the original multiscale problem, using the FE solutions of the multiscale homogenized problem. In the case of two scales, we derive an explicit error estimate in terms of the homogenization error and the FE error. It is well known that for two-scale elliptic problems in a domain $$D$$, if the solution of the homogenized problem belongs to $$H^2(D)$$, and the solutions to the cell problems are sufficiently smooth, the homogenization error in the $$H^1(D)$$ norm is $${\mathcal O}(\varepsilon^{1/2}),$$ where $$\varepsilon$$ is the microscopic scale (Bensoussan et al., 1978; Jikov et al., 1994). For two-scale Maxwell-type equations, the $${\mathcal O}(\varepsilon^{1/2})$$ homogenization error in the $$H({\rm curl},D)$$ norm is obtained when the solution $$u_0$$ of the homogenized problem (4.7) belongs to $$H^1({\rm curl},D)$$. However, for polygonal domains that are of interest in FE discretization, $$u_0$$ generally belongs only to a weaker regularity space $$H^s({\rm curl\,},D)$$ for $$0<s<1$$ (see e.g., Hiptmair, 2002). For this case, we develop an approach to deriving a new homogenization error estimate. Though we present the result for Maxwell-type equations, the approach works verbatim for two-scale elliptic and elasticity problems when the solution to the homogenized problem is in $$H^{1+s}(D)$$. As far as we are aware, this is a new result in the homogenization theory and forms another main contribution of the article. For the case of more than two scales, an analytic homogenization error is not available. However, we can still derive a corrector from the FE solution of the multiscale homogenized problem, albeit without an explicit rate of convergence. This article is organized as follows. In the next section, we formulate the multiscale Maxwell-type equation. Homogenization of the multiscale Maxwell-type equation (2.4) is studied in Bensoussan et al. (1978) in the two-scale case, using two-scale asymptotic expansion. Here, we use the multiscale convergence method to study (2.4) in the general multiscale setting. We thus develop multiscale convergence for a bounded sequence in $$H({\rm curl\,},D)$$. Two-scale convergence for a bounded sequence in $$H({\rm curl\,},D)$$ is developed in Wellander & Kristensson (2003). Since we consider the general $$(n+1)$$-scale convergence and the limiting result that we use is in a slightly different form from that of Wellander & Kristensson (2003) in the two-scale case, so we present the proofs in full. FE approximations of the multiscale homogenized Maxwell-type problem are studied in Section 3. We prove the FE error estimates in cases of both full and sparse tensor product FE approximations. The errors are essentially equal (apart from a logarithmic multiplying factor), but the dimension of the sparse tensor product FE space is much lower than that of the full tensor product FE space and is essentially equal to that for solving a problem in $$\mathbb{R}^d$$ only. In Section 4, we construct numerical correctors for the solution to the original multiscale problem. For two-scale problems, we prove the general homogenization error estimate for the case where $$u_0$$ belongs to the weaker regularity space $$H^s({\rm curl\,},D),$$ where $$0<s<1$$. From that we deduce the error estimate for the numerical corrector, which is of the order of the sum of the homogenization error estimate and the FE error. For the case of more than two scales, we derive a numerical corrector but without a rate of convergence. In Section 5, we prove that the regularity required to get the FE error estimate for the sparse tensor product FEs and to get the homogenization error in the two-scale case is achievable. Section 6 contains numerical experiments that confirm our analysis. Finally, the two Appendices A and B contain the long proofs of some previous results: the proof of the homogenization error when $$u_0$$ belongs to a weaker regularity space is presented in Appendix A. Throughout the article, by $$\#$$ we denote the spaces of functions that are periodic with the period being the unit cube $$Y\subset \mathbb{R}^d$$. Repeated indices indicate summation. The notations $$\nabla$$ and $${\rm curl\,}$$ without indicating the variable explicitly denote the gradient and the $${\rm curl\,}$$ operator with respect to $$x$$ of a function of $$x$$ only, where $$\nabla_x$$ and $${\rm curl\,}_{\!x}$$ denote the partial gradient and partial $${\rm curl\,}$$ of a function depending on $$x$$ and also on other variables. We generally present the theoretical results for the three-dimensional case and mention the two-dimensional case only when it is necessary, as the two cases are largely similar. 2. Problem setting 2.1 Multiscale Maxwell-type problems Let $$D$$ be a domain in $$\mathbb{R}^d$$ ($$d=2,3$$). Let $$Y$$ be the unit cube in $$\mathbb{R}^d$$. By $$Y_1,\ldots,Y_n$$ we denote $$n$$ copies1 of $$Y$$. We denote by $${\bf Y}$$ the product set $$Y_1\times Y_2\times\cdots\times Y_n$$ and by $$\boldsymbol{y}\in{\bf Y}$$ the vector $$\boldsymbol{y}=(y_1,y_2,\ldots,y_n)$$. For each $$i=1,\ldots,n$$, we denote by $${\bf Y}_i$$ the set of vectors $$\boldsymbol{y}_i=(y_1,\ldots,y_i),$$ where $$y_j\in Y_j$$ for $$j=1,\ldots,i$$. For $$d=3$$, let $$a$$ and $$b$$ be functions with symmetric matrix values from $$D\times {\bf Y}$$ to $$\mathbb{R}^{d\times d}_{\rm sym}$$; $$a$$ and $$b$$ are continuous in $$D\times {\bf Y}$$ and are periodic with respect to each variable $$y_i$$ with the period being $$Y_i$$. We assume that for all $$x\in D$$ and $$\boldsymbol{y}\in{\bf Y}$$, and all $$\xi,\zeta\in \mathbb{R}^d$$,   c∗|ξ|2≤aij(x,y)ξiξj,   aij(x,y)ξiζj≤c∗|ξ||ζ|,c∗|ξ|2≤bij(x,y)ξiξj,   bij(x,y)ξiζj≤c∗|ξ||ζ|, (2.1) where $$c_*$$ and $$c^*$$ are positive numbers; $$|\cdot|$$ denotes the Euclidean norm in $$\mathbb{R}^3$$. Let $$\varepsilon$$ be a small positive value, and $$\varepsilon_1,\ldots,\varepsilon_n$$ be $$n$$ functions of $$\varepsilon$$ that denote the $$n$$ microscopic scales that the problem depends on. We assume the following scale separation properties: for all $$i=1,\ldots,n-1$$,   limε→0εi+1(ε)εi(ε)=0. (2.2) Without loss of generality, we assume that $$\varepsilon_1=\varepsilon$$. We define $$a^\varepsilon, b^\varepsilon: D\to\mathbb{R}^{d\times d}_{\rm sym}$$ as   aε(x)=a(x,xε1,…,xεn),  bε(x)=b(x,xε1,…,xεn). (2.3) Let   W=H0(curl,D)={u∈L2(D)3,  curlu∈L2(D)3,  u×ν=0}, where $$\nu$$ denotes the outward normal vector on the boundary $$\partial D$$. Let $$f\in W'$$. We consider the problem   curl(aε(x)curluε(x))+bε(x)uε(x)=f(x), (2.4) with the boundary condition $$u^{\varepsilon}\times \nu=0$$ on $$\partial D$$. We formulate this problem in the variational form as follows: find $$u^{\varepsilon}\in W$$ so that   ∫D[aε(x)curluε(x)⋅curlϕ(x)+bε(x)uε(x)⋅ϕ(x)]dx=∫Df(x)⋅ϕ(x)dx (2.5) for all $$\phi\in W$$ (by $$\int_D\,f\cdot\phi\, {\rm d}x$$ we denote the duality pairing between $$W'$$ and $$W$$). The Lax–Milgram lemma guarantees the existence of a unique solution $$u^{\varepsilon}$$ that satisfies   ‖uε‖W≤c‖f‖W′, (2.6) where the constant $$c$$ depends only on $$c_*$$ and $$c^*$$ in (2.1). For $$d=2$$, the matrix function $$b^\varepsilon:D\times{\bf Y}\to \mathbb{R}^{2\times 2}$$ is defined as above. As $${\rm curl\,}u^{\varepsilon}$$ is now a scalar function, $$a(x,\boldsymbol{y})$$ is a continuous function from $$D\times{\bf Y}$$ to $$\mathbb{R},$$ which is periodic with respect to each variable $$y_i$$ with the period being $$Y_i$$. In the place of (2.1), we have   c∗≤a(x,y)≤c∗  ∀x∈D and y∈Y. The variational formulation in two dimensions becomes   ∫D[aε(x)curluε(x)curlϕ(x)+bε(x)uε(x)⋅ϕ(x)]dx=∫Df(x)⋅ϕ(x)dx  ∀ϕ∈W. (2.7) In the rest of the article, we present the results for the three-dimensional case and only mention the two-dimensional case when necessary; the results for two dimensions are similar. 2.2 Multiscale convergence We use multiscale convergence to derive the homogenized equation. We first recall the definition of multiscale convergence (see Nguetseng, 1989; Allaire, 1992; Allaire & Briane, 1996). Definition 2.1 A sequence of functions $$\{w^\varepsilon\}_\varepsilon\subset L^2(D)$$$$(n+1)$$-scale converges to a function $$w^0\in L^2(D\times {\bf Y})$$ if for all smooth functions $$\phi\in C^\infty(D\times{\bf Y}),$$ which are periodic with respect to $$y_i$$ with the period being $$Y_i$$ for $$i=1,\ldots,n$$,   limε→0∫Dwε(x)ϕ(x,xε1,…,xεn)dx=∫D∫Yw0(x,y)ϕ(x,y)dydx. We have the following result. Proposition 2.2 From a bounded sequence in $$L^2(D),$$ we can extract an $$(n+1)$$-scale convergent subsequence. For a bounded sequence in $$H({\rm curl\,},D)$$, we have the following results on $$(n+1)$$-scale convergence. These results were first established in Wellander & Kristensson (2003) for the two-scale case. We present below the multiscale convergence of a bounded sequence in $$H({\rm curl\,},D),$$ which will be used to study the multiscale equations (2.5) and (2.7). By $$\tilde H_\#({\rm curl\,},Y_i)$$ we denote the equivalent classes of functions in $$H_\#({\rm curl\,},Y_i)$$ such that if $${\rm curl\,} v={\rm curl\,} w$$ we regard $$v=w$$ in $$\tilde H_\#({\rm curl\,},Y_i)$$. Proposition 2.3 Let $$\{w^\varepsilon\}_\varepsilon$$ be a bounded sequence in $$H({\rm curl\,},D)$$. There is a subsequence (not renumbered), a function $$w_0\in H({\rm curl\,},D)$$, $$n$$ functions $${\frak w_i}\in L^2(D\times Y_1\times\cdots\times Y_{i-1},H^1_\#(Y_i)/\mathbb{R})$$ such that   wε⟶(n+1)−scalew0+∑i=1n∇yiwi. Further, there are $$n$$ functions $$w_i\in L^2(D\times Y_1\times\cdots\times Y_{i-1},\tilde H_\#({\rm curl\,},Y_i))$$ such that   curlwε⟶(n+1)−scalecurlw0+∑i=1ncurlyiwi. Proof. Let $$\xi\in L^2(D\times{\bf Y})^3$$ be the $$(n+1)$$-scale limit of $$\{w^\varepsilon\}_\varepsilon$$. Consider the function $$\phi=\varepsilon_n{\it{\Phi}}(x,y_1,\ldots,y_n),$$ where $${\it{\Phi}}$$ is a function in $$C^\infty_0(D,C^\infty_\#(Y_1,\ldots,C^\infty_\#(Y_n),\ldots))^3$$ and is periodic with respect to $$y_1, \ldots,y_n$$ with the period being $$Y_1,\ldots,Y_n,$$ respectively. We then have   limε→0∫Dcurlwε⋅εnΦ(x,xε1,…,xεn)dx=0. On the other hand,   limε→0∫Dcurlwε⋅εnΦ(x,xε1,…,xεn)dx = limε→0∫Dwε⋅εncurlΦ(x,xε1,…,xεn)dx = limε→0∫Dwε⋅curlynΦ(x,xε1,…,xεn)dx = ∫D∫Yξ(x,y)⋅curlynΦ(x,y)dydx. Thus, there is a function $$\xi_{n-1}(x,\boldsymbol{y}_{n-1})\in L^2(D\times{\bf Y}_{n-1})$$ and a function $${\frak w_n}(x,\boldsymbol{y}_n)\in L^2(D\times{\bf Y}_{n-1},H^1_\#(Y_n)/\mathbb{R})$$ such that   ξ(x,y)=ξn−1(x,yn−1)+∇ynwn(x,y). Next we choose $$\phi=\varepsilon_{n-1}{\it{\Phi}}(x,y_1,\ldots,y_{n-1})$$ for a function $${\it{\Phi}}\in C^\infty_0(D,C^\infty_\#(Y_1,\ldots,C^\infty_\#(Y_{n-1}),\ldots),$$ which is periodic with respect to $$y_1,\ldots,y_{n-1}$$. We then have   0 = limε→0∫Dcurlwε⋅εn−1Φ(x,xε1,…,xεn−1)=limε→0∫Dwε⋅curlyn−1Φ(x,xε1,…,xεn−1)dx = ∫D∫Y(ξn−1(x,yn−1)+∇ynwn(x,y))⋅curlyn−1Φ(x,y1,…,yn−1)dyn−1dx = ∫D∫Yn−1ξn−1(x,yn−1)⋅curlyn−1Φ(x,y1,…,yn−1)dyn−1dx. From this, there is a function $$\xi_{n-2}(x,\boldsymbol{y}_{n-2})\in L^2(D\times {\bf Y}_{n-2})$$ and a function $${\frak w_{n-1}}(x,\boldsymbol{y}_{n-1})\in L^2(D\times {\bf Y}_{n-2},H^1_\#(Y_{n-1})/\mathbb{R})$$ so that   ξn−1(x,yn−1)=ξn−2(x,yn−2)+∇yn−1wn−1(x,yn−1), so   ξ(x,y)=ξn−2(x,yn−2)+∇yn−1wn−1(x,yn−1)+∇ynwn(x,y). Continuing this process, we have   ξ(x,y)=w0(x)+∑i=1n∇yiwi(x,yi), where $$w_0\in L^2(D)^3$$ and $${\frak w_i}(x,\boldsymbol{y}_i)\in L^2(D\times {\bf Y}_{i-1},H^1_\#(Y_i))$$. As $$\int_Y\xi(x,\boldsymbol{y})\,{\rm d}\boldsymbol{y}=w_0(x)$$, $$w_0$$ is the weak limit of $$w^\varepsilon$$ in $$L^2(D)^3$$. Let $$\eta(x,\boldsymbol{y})$$ be the $$(n+1)$$-scale convergence limit of $${\rm curl\,} w^\varepsilon$$ in $$L^2(D\times{\bf Y})$$. Let $${\it{\Phi}}(x,y_1,\ldots,y_n)\in C^\infty_0(D,C^\infty_\#(Y_1,\ldots,C^\infty_\#(Y_n),\ldots))$$. We have    ∫Dcurlwε⋅∇Φ(x,xε1,…,xεn)dx  =∫Dwε⋅curl∇Φ(x,xε1,…,xεn)dx−∫∂D(wε×ν)⋅∇Φ(x,xε1,…,xεn)ds=0. Thus,   0 = limε→ 0∫Dcurlwε⋅εn∇Φ(x,xε1,…,xεn)dx=limε→0∫Dcurlwε⋅∇ynΦ(x,xε1,…,xεn)dx = ∫D∫Yη(x,y)⋅∇ynΦ(x,y1,…,yn)dydx. Therefore, there is a function $$w_n(x,\boldsymbol{y}_n)\in L^2(D\times{\bf Y}_{n-1},\tilde H_\#({\rm curl\,},Y_n))$$ and a function $$\eta_{n-1}(x,\boldsymbol{y}_{n-1})\in L^2(D\times{\bf Y}_{n-1})$$ such that   η(x,y)=ηn−1(x,yn−1)+curlynwn(x,y). Let $${\it{\Phi}}(x,y_1,\ldots,y_{n-1})\in C^\infty_0(D,C^\infty_\#(Y_1,\ldots,C^\infty_\#(Y_{n-1}),\ldots))$$. We have   0 = limε→ 0∫Dcurlwε⋅εn−1∇Φ(x,xε1,…,xεn−1)dx=limε→0∫Dcurlwε⋅∇yn−1Φ(x,xε1,…,xεn−1)dx = ∫D∫Y(ηn−1(x,yn−1)+curlynwn(x,y))⋅∇yn−1Φ(x,y1,…,yn−1)dydx = ∫D∫Yn−1ηn−1(x,yn−1)⋅∇yn−1ϕ(x,y1,…,yn−1)dyn−1dx. Therefore, there is a function $$w_{n-1}(x,\boldsymbol{y}_{n-1})\in L^2(D\times{\bf Y}_{n-2},\tilde H_\#({\rm curl\,},Y_{n-1}))$$ and a function $$\eta_{n-2}(x,\boldsymbol{y}_{n-2})\in L^2(D\times {\bf Y}_{n-2})^3$$ so that   ηn−1(x,yn−1)=ηn−2(x,yn−2)+curlyn−1wn−1(x,yn−1) so   η(x,y)=ηn−2(x,yn−2)+curlyn−1wn−1(x,yn−1)+curlynwn(x,y). Continuing, we find that there is a function $$\eta_0(x)\in L^2(D)^3$$ and functions $$w_i(x,\boldsymbol{y}_{i})\in L^2(D\times{\bf Y}_{i-1},\tilde H_\#({\rm curl\,},Y_i))$$ so that   η(x,y)=η0(x)+∑i=1ncurlyiwi(x,yi). As for all $$\phi(x)\in C^\infty_0(D)^3$$  limε→0∫Dcurlwε(x)⋅ϕ(x)dx=∫Dη0(x)⋅ϕ(x)dx,$$\eta_0$$ is the weak limit of $${\rm curl\,} w^\varepsilon$$ in $$L^2(D)^3$$. Thus, $$\eta_0={\rm curl\,} w_0$$. We then get the conclusion. □ 2.3 Multiscale homogenized Maxwell-type problem From (2.6) and Proposition 2.3, we can extract a subsequence (not renumbered), a function $$u_0\in H_0({\rm curl\,}, D)$$, $$n$$ functions $${\frak u_i}\in L^2(D\times Y_1\times\cdots\times Y_{i-1},H^1_\#(Y_i)/\mathbb{R})$$ and $$n$$ functions $$u_i\in L^2(D\times Y_1\times\cdots\times Y_{i-1},\tilde H_\#({\rm curl\,},Y_i))$$ such that   uε⟶(n+1)−scaleu0+∑i=1n∇yiui (2.8) and   curluε⟶(n+1)−scalecurlu0+∑i=1ncurlyiui. (2.9) For $$i=1,\ldots,n$$, let $$W_i=L^2(D\times Y_1\times\cdots\times Y_{i-1},\tilde H_\#({\rm curl\,},Y_i))$$ and $$V_i=L^2(D\times Y_1\times\cdots\times Y_{i-1},H^1_\#(Y_i)/\mathbb{R})$$. We define the space $${\bf V}$$ as   V=W×W1×⋯×Wn×V1×⋯×Vn. For $$\boldsymbol{v}=(v_0,\{v_i\},\{\frak v_i\})\in {\bf V}$$, we define the norm   |||v|||=‖v0‖H(curl,D)+∑i=1n‖vi‖L2(D×Yi−1,H~#(curl,Yi))+∑i=1n‖vi‖L2(D×Yi−1,H#1(Yi)/R). We then have the following result. Proposition 2.4 We define $$\boldsymbol{u}=(u_0,\{u_i\}, \{\frak u_i\})\in{\bf V}$$. Then $$\boldsymbol{u}$$ satisfies   B(u,v):=∫D∫Y[a(x,y)(curlu0+∑i=1ncurlyiui)⋅(curlv0+∑i=1ncurlyivi)  +b(x,y) (u0+∑i=1n∇yiui)⋅(v0+∑i=1n∇yivi)]dydx=∫Df(x)⋅v0(x)dx (2.10) for all $$\boldsymbol{v}=(v_0,\{v_i\},\{\frak v_i\})\in {\bf V}$$. Proof. Let $$v_0\in C^\infty_0(D)^3$$, $$v_i\in C^\infty_0(D, C^\infty_\#(Y_1,\ldots,C^\infty_\#(Y_i),\ldots))^3$$ and $${\frak v_i}\in C^\infty_0(D,C^\infty_\#(Y_1,\ldots,C^\infty_\# (Y_i),\ldots))$$ for $$i=1,\ldots,n$$. Let the test function $$v$$ in (2.5) be   v(x)=v0(x)+∑i=1nεi(vi(x,xε1,…,xεi)+∇vi(x,xε1,…,xεi)). We have    ∫D[aε(x)curluε(x)⋅(curlv0(x)+∑i=1nεicurlxvi(x,xε1,…,xεi)+∑i=1n∑j=1iεiεjcurlyjvi(x,xε1,…,xεi)   +  ∑i=1nεicurl∇vi(x,xε1,…,xεi))  +bε(x)uε(x)⋅(v0(x)+∑i=1nεivi(x,xε1,…,xεi)   +∑i=1nεi∇xvi(x,xε1,…,xεi)+∑i=1n∑j=1iεiεj∇yjvi(x,xε1,…,xεi))]dx  =∫Df(x)⋅(v0(x)+∑i=1nεivi(x,xε1,…,xεi)  +∑i=1nεi∇xvi(x,xε1,…,xεn)+∑i=1n∑j=1iεiεj∇yjvi(x,xε1,…,xεi)). Using multiscale convergence and the scale separation (2.2), letting $$\varepsilon$$ go to 0, we have    ∫D∫Y[a(x,y)(curlu0+∑i=1ncurlyiui)⋅(curlv0+∑i=1ncurlyivi)  +b(x,y)(u0+∑i=1n∇yiui)⋅(v0+∑i=1n∇yivi)]dydx =∫Df(x)⋅v0(x)dx+∫D∫Yf(x)⋅∑i=1n∇yivi(x,y1,…,yi)dydx =∫Df(x)⋅v0(x)dx. Using a density argument, we have (2.10). □ Proposition 2.5 The bilinear form $$B:{\bf V}\times{\bf V}\to \mathbb{R}$$ is coercive and bounded, i.e., there are positive constants $$C^*$$ and $$C_*$$ so that   B(u,v)≤C∗|||u||||||v|||andC∗|||u||||||u|||≤B(u,u) (2.11) for all $$\boldsymbol{u},\boldsymbol{v}\in {\bf V}$$. Problem (2.10) thus has a unique solution. The convergence relations (2.8) and (2.9) hold for the whole sequence $$\{u^{\varepsilon}\}_\varepsilon$$. Proof. It is easy to see that there is a positive constant $$C^*$$ such that   B(u,v)≤C∗|||u||||||v|||. Now we show that $$B$$ is coercive. We have from (2.1),   B(u,u) ≥c∗∫D∫Y(|curlu0+∑i=1ncurlyiui|2+|u0+∑i=1n∇yiui|2)dydx ≥c∫D∫Y(|curlu0|2+∑i=1n|curlyiui|2+|u0|2+|∇yiui|2)dydx≥c|||u|||2. We then get the conclusion from Lax–Milgram lemma. □ 3. FE discretization Let $$D$$ be a polygonal domain in $$\mathbb{R}^3$$. We consider a hierarchy of simplices $$\mathcal{T}^l$$ ($$l=0,1,\ldots$$), where $$\mathcal{T}^{l+1}$$ is obtained from $$\mathcal{T}^l$$ by dividing each simplex in $$\mathcal{T}^l$$ into eight tedrahedra. The mesh size of $$\mathcal{T}^l$$ is $$h_l={\mathcal O}(2^{-l})$$. For each tedrahedron $$T$$, we consider the edge FE space   R(T)={v:  v=α+β×x,  α,β∈R3}. When $$d=2$$, $$\mathcal{T}^{l+1}$$ is obtained from $$\mathcal{T}^l$$ by dividing each simplex in $$\mathcal{T}^l$$ into four congruent triangles. For each triangle $$T$$, we consider the edge FE space   R(T)={v:  v=(α1α2)+β(x2−x1)}, where $$\alpha_1,\alpha_2$$ and $$\beta$$ are constants. Alternatively, if the domain can be partitioned into a set of cubes, we can use edge FE on a cubic mesh instead (see Monk, 2003). We denote by $$\mathcal{P}_1(T)$$ the set of linear polynomials in each simplex $$T$$. In the following, we present the analysis for the three-dimensional case only; the two-dimensional case is similar. For the cube $$Y$$, we partition it into a hierarchy of simplices $$\mathcal{T}^l_\#,$$ which are distributed periodically. We consider the FE spaces   Wl ={v∈H0(curl,D), v|T∈R(T) ∀T∈Tl},Vl ={v∈H1(D), v|T∈P1(T) ∀T∈Tl},W#l ={v∈H#(curl,Y), v|T∈R(T) ∀T∈T#l} and   V#l={v∈H#1(Y), v|T∈P1(T) ∀T∈T#l}. For $$d=2,3$$, we have the following estimates (see Ciarlet, 1978; Monk, 2003):   infvl∈Wl‖v−vl‖H(curl,D)≤chls(‖v‖Hs(D)d+‖curlv‖Hs(D)d) for all $$v\in H_0({\rm curl\,}, D)\bigcap H^s({\rm curl\,},D)$$;   infvl∈W#l‖v−vl‖H#(curl,Y)≤chls(‖v‖Hs(Y)d+‖curlv‖Hs(Y)d) for all $$v\in H_\#({\rm curl\,}, Y)\bigcap H^s({\rm curl\,}, Y)$$;   infvl∈Vl‖v−vl‖L2(D)≤chls‖v‖Hs(D) for all $$v\in H^s(D)$$;   infvl∈V#l‖v−vl‖L2(Y)≤chls‖v‖Hs(Y) for all $$v\in H^s_\#(Y)$$ and   infvl∈V#l‖v−vl‖H#1(Y)≤chls‖v‖H1+s(Y) for all $$v\in H^1_\#(Y)\bigcap H^{1+s}(Y)$$. 3.1 Full tensor product FEs As $$L^2(D\times{\bf Y}_{i-1},\tilde H_\#({\rm curl\,},Y_i))\cong L^2(D)\otimes L^2(Y_1)\otimes\cdots\otimes L^2(Y_{i-1})\otimes \tilde H_\#({\rm curl\,},Y_i),$$ we use the tensor product FE space   Wil=Vl⊗V#l⊗⋯⊗V#l⏟i−1 times ⊗W#l to approximate $$u_i$$. Similarly, as $${\frak u_i}\in L^2(D\times{\bf Y}_{i-1},H^1_\#(Y))$$, we use the FE space   Vil=Vl⊗V#l⊗⋯⊗V#l⏟i times  to approximate $${\frak u_i}$$. We define the space   Vl=Wl×W1l×⋯×Wnl×V1l×⋯×Vnl. The full tensor product FE approximating problem is, find $$\boldsymbol{u}^L\in{\bf V}^L$$ so that   B(uL,vL)=∫Df(x)⋅v0L(x)dx  ∀vL=(v0L,{viL},viL)∈VL. (3.1) To get an error estimate for this FE approximating problem, we define the following regularity spaces for $${\frak u_i}$$ and $$u_i$$. For the functions $$u_i$$, we define the regularity space $$\mathcal{H}_i$$ of functions $$w$$ in $$L^2(D\times Y_1\times\cdots\times Y_{i-1},H^1_\#({\rm curl\,},Y_i))$$ such that for all $$k=1,2,3$$,   ∂w∂xk∈L2(D×Y1×⋯×Yi−1,H~#(curl,Yi)) and for all $$j=1,\ldots,i-1$$ and $$k=1,2,3$$,   ∂w∂(yj)k∈L2(D×Y1×⋯×Yi−1,H~#(curl,Yi)). In other words, for all $$w\in \mathcal{H}_i$$, $$w$$ belongs to $$L^2(D\times Y_1\times\cdots\times Y_{i-1},H^1_\#({\rm curl\,},Y_i))$$, $$L^2(Y_1\times\cdots\times Y_{i-1},H^1(D,\tilde H_\#({\rm curl\,},Y_i)))$$ and $$L^2(D\times\prod_{k<i,k\ne j}Y_k,H^1_\#(Y_j,\tilde H_\#({\rm curl\,},Y_i)))$$ for $$j=1,\ldots,i-1$$. For $$0<s<1$$, we define the space $$\mathcal{H}^s_i$$ by interpolation. It consists of functions $$w$$ such that $$w$$ belongs to $$L^2(D\times Y_1\times\cdots\times Y_{i-1},H^s_\#({\rm curl\,},Y_i))$$, $$L^2(Y_1\times\cdots\times Y_{i-1},H^s(D,\tilde H_\#({\rm curl\,},Y_i)))$$ and $$L^2(D\times\prod_{k<i,k\ne j}Y_k,H^s_\#(Y_j,\tilde H_\#({\rm curl\,},Y_i)))$$. We equip $$\mathcal{H}_i^s$$ with the norm   ‖w‖His =‖w‖L2(D×Y1×⋯×Yi−1,H#s(curl,Yi))+‖w‖L2(Y1×⋯×Yi−1,Hs(D,H~#(curl,Yi)))  +∑j=1i−1‖w‖L2(D×∏k<i,k≠j,H#s(Yj,H~#(curl,Yi))). We then have the following lemma. Lemma 3.1 For $$w\in \mathcal{H}_i^s$$,   infwl∈Wil‖w−wl‖L2(D×Y1×⋯×Yi−1,H~#(curl,Yi))≤chls‖w‖His. The proof of this lemma is similar to that for full tensor product FEs in Hoang & Schwab (2004/05) and Bungartz & Griebel (2004), using orthogonal projection. We refer to Hoang & Schwab (2004/05) and Bungartz & Griebel (2004) for details. We define $${\frak H_i}^s$$ as the space of functions $$w\in L^2(D\times Y_1\times\cdots\times Y_{i-1},H^{1+s}_\#(Y_i))$$ such that $$w\in L^2(Y_1\times\cdots\times Y_{i-1},H^s(D,H^1_\#(Y_i)))$$ and for all $$j=1,\ldots,i-1$$, $$ w\in L^2(D\times\prod_{k<i,k\ne j}Y_k,H^s_\#(Y_{j},H^1_\#(Y_i)))$$. We then define the norm   ‖w‖His =‖w‖L2(D×Y1×⋯×Yi−1,H#1+s(Yi))+‖w‖L2(Y1×⋯×Yi−1,Hs(D,H1(Yi)))  +∑j=1i−1‖w‖L2(D×∏k<i,k≠jYk,Hs(Yj,H1(Yi))). We have the following result. Lemma 3.2 For $$w\in {\frak H_i}^s$$,   infwl∈Vil‖w−wl‖L2(D×Y1×⋯×Yi−1,H#1(Yi))≤chls‖w‖His. We then define the regularity space   Hs=Hs(curl,D)×H1s×⋯×Hns×H1s×⋯×Hns with the norm   ‖w‖Hs=‖w0‖Hs(curl,D)+∑i=1n‖wi‖His+∑i=1n‖wi‖His for $$\boldsymbol{w}=(w_0,\{w_i\},\{\frak w_i\})\in \boldsymbol{\mathcal{H}}^s$$. We have the following approximation result. Lemma 3.3 For $$\boldsymbol{w}\in \boldsymbol{\mathcal{H}}^s$$  infwl∈Vl‖w−wl‖V≤chls‖w‖Hs. From the boundedness and coerciveness conditions (2.11), using Cea’s lemma, we deduce the following result. Proposition 3.4 If $$\boldsymbol{u}\in \boldsymbol{\mathcal{H}}^s$$, for the full tensor product FE approximating problem (3.1) we have the error estimate   ‖u−uL‖V≤chLs‖u‖Hs. (3.2) 3.2 Sparse tensor product FEs We define the following orthogonal projection:   Pl0:L2(D)→Vl,P#l0:L2(Y)→V#l with the convention $$P^{-10}=0$$, $$P^{-10}_\#=0$$. We define the following detail spaces:   Vl=(Pl0−P(l−1)0)Vl,  V#l=(P#l0−P#(l−1)0)Vl. Since   Vl=⨁0≤i≤lViandV#l=⨁0≤i≤lV#i, the full tensor product spaces $$W_i^L$$ and $$V_i^L$$ are defined as   WiL=(⨁0≤l0,…,li−1≤LVl0⊗V#l1⊗⋯⊗V#li−1)⊗W#L and   ViL=(⨁0≤l0,…,li−1≤LVl0⊗V#l1⊗⋯⊗V#li−1)⊗V#L. We then define the sparse tensor product FE spaces as   W^iL=⨁l0+⋯+li−1≤LVl0⊗V#l1⊗⋯⊗V#li−1⊗W#L−(l0+⋯+li−1) and   V^iL=⨁l0+⋯+li−1≤LVl0⊗V#l1⊗⋯⊗V#li−1⊗V#L−(l0+⋯+li−1). The function $$\boldsymbol{u}$$ is approximated by the space   V^L=WL⊗W^1L⊗⋯⊗W^nL⊗V^1L⊗⋯⊗V^nL. The sparse tensor product FE approximating problem is, find $$\widehat {\bf{u}}^L\in \hat{\bf V}^L$$ such that   B(u^L,v^L)=∫Df(x)⋅v^0L(x)dx  ∀v^L=(v^0L,{v^iL},{v^iL})∈V^L. (3.3) From the coerciveness and boundedness conditions in (2.11), using Cea’s lemma we deduce the error estimate for the sparse tensor product approximating problem   ‖u−u^L‖V≤cinfv^L∈VL‖u−v^L‖V. To quantify the error estimate, we use the following regularity spaces. We define $$\hat{\mathcal{H}}_i$$ as the space of functions $$w\in L^2(D\times Y_1\times\cdots\times Y_{i-1},H^1_\#({\rm curl\,},Y_i)),$$ which are periodic with respect to $$y_j$$ with the period being $$Y_j$$ ($$j=1,\ldots,i-1$$) such that for any $$\alpha_0,\alpha_1,\ldots,\alpha_{i-1}\in \mathbb{N}_0^d$$ with $$|\alpha_k|\le 1$$ for $$k=0,\ldots,i-1$$,   ∂|α0|+|α1|+⋯+|αi−1|∂xα0∂y1α1⋯∂yi−1αi−1w∈L2(D×Y1×⋯×Yi−1,H#1(curl,Yi)). We equip $$\hat{\mathcal{H}}_i$$ with the norm   ‖w‖H^i=∑αj∈Rd,|αj|≤10≤j≤i−1‖∂|α0|+|α1|+⋯+|αi−1|∂xα0∂y1α1⋯∂yi−1αi−1w‖L2(D×Y1×⋯×Yi−1,H#1(curl,Yi)). We can write $$\hat{\mathcal{H}}_i$$ as $$H^1(D,H^1_\#(Y_1,\ldots,H^1_\#(Y_{i-1},H^1_\#({\rm curl\,},Y_i)),\ldots))$$. By interpolation, we define $$\hat{\mathcal{H}}_i^s=H^s(D,H^s_\#(Y_1,\ldots,H^s_\#(Y_{i-1},H^s_\#({\rm curl\,},Y_i)),\ldots))$$ for $$0<s<1$$. We define $$\hat{\frak H}_i$$ as the space of functions $$w\in L^2(D\times Y_1\times\cdots\times Y_{i-1},H^2_\#(Y_i))$$ that are periodic with respect to $$y_j$$ with the period being $$Y_j$$ for $$j=1,\ldots,i-1$$ such that $$\alpha_0,\alpha_1,\ldots,\alpha_{i-1}\in \mathbb{N}_0^d$$ with $$|\alpha_k|\le 1$$ for $$k=0,\ldots,i-1$$,   ∂|α0|+|α1|+⋯+|αi−1|∂xα0∂y1α1⋯∂yi−1αi−1w∈L2(D×Y1×⋯×Yi−1,H#2(Yi)). The space $$\hat{\frak H}_i$$ is equipped with the norm   ‖w‖H^i=∑αj∈Rd,|αj|≤10≤j≤i−1‖∂|α0|+|α1|+⋯+|αi−1|∂xα0∂y1α1⋯∂yi−1αi−1w‖L2(D×Y1×⋯×Yi−1,H#2(Yi)). We can write $$\hat{\frak H}_i$$ as $$H^1(D,H^1_\#(Y_1,\ldots,H^1_\#(Y_{i-1},H^2_\#(Y_i))))$$. By interpolation, we define the space $$\hat{\frak H}_i^s:=H^s(D,H^s(Y_1,\ldots,H^s(Y_{i-1},H^{1+s}_\#(Y_i))))$$. The regularity space $$\hat{\boldsymbol{\mathcal{H}}}^s$$ is defined as   H^s=Hs(curl,D)×H^1s×⋯H^ns×H^1s×⋯×H^ns. Lemmas 3.5 and 3.6 present the approximating properties of functions in $$\hat{\mathcal{H}}_i^s$$ and $${\hat{\frak H}_i^s}$$. The proofs follow from those for sparse tensor products in Hoang & Schwab (2004/05) and Bungartz & Griebel (2004). Lemma 3.5 For $$w\in \hat{\mathcal{H}}_i^s$$,   infwL∈W^iL‖w−wL‖L2(D×Y1×⋯×Yi−1,H#(curl,Yi))≤cLi/2hLs‖w‖H^is. Similarly we have the following lemma. Lemma 3.6 For $$w\in \hat{\frak H}_i^s$$,   infwL∈V^iL‖w−wL‖L2(D×Y1×⋯×Yi−1,H#1(Yi))≤cLi/2hLs‖w‖H^is. From these lemmas we deduce the following result. Lemma 3.7 For $$\boldsymbol{w}\in \hat{\boldsymbol{\mathcal{H}}}^s$$,   infwL∈V^L‖w−wL‖V≤cLn/2hLs‖w‖H^s. From this we deduce the following error estimate for the sparse tensor product FE problem (3.3). Proposition 3.8 If the solution $$\boldsymbol{u}$$ of problem (2.10) belongs to $$\hat{\boldsymbol{\mathcal{H}}}^s$$ then   ‖u−u^‖V≤cLn/2hLs‖u‖H^s. Remark 3.9 The dimension of the full tensor product FE space $${\bf V}^L$$ is $${\mathcal O}(2^{dnL}),$$ which is very large when $$L$$ is large. The dimension of the sparse tensor product FE space $$\hat{\bf V}^L$$ is $${\mathcal O}(L^n2^{dL}),$$ which is essentially equal to the number of degrees of freedom for solving a problem in $$\mathbb{R}^d$$ obtaining the same level of accuracy. 4. Convergence in physical variables We employ the FE solutions for the multiscale homogenized Maxwell-type equation (2.10) in the previous section to derive numerical correctors for the solution $$u^{\varepsilon}$$ of the multiscale problem (2.4). In the two-scale case, we derive the homogenization error explicitly in terms of $$\varepsilon$$ so that an error in terms of the microscopic scale $$\varepsilon$$ and the mesh size is obtained for the numerical corrector. We consider the general case, where the solution $$u^0$$ of the homogenized problem belongs to the space $$H^s({\rm curl\,},D)$$ for $$0<s\le 1$$, thus generalizing the standard homogenization rate of convergence $$\varepsilon^{1/2}$$ for elliptic problems (see e.g, Bensoussan et al., 1978; Jikov et al., 1994). This is a new result in homogenization theory. We present it for two-scale Maxwell-type equations, but the procedure works verbatim for two-scale elliptic and elasticity problems, where the solutions of the homogenized problems belong to $$H^{1+s}(D)$$. We present this section for the case $$d=3$$; the case $$d=2$$ is similar. 4.1 Two-scale problems For the two-scale case, we denote the function $$a(x,\boldsymbol{y})$$ by $$a(x,y)$$. The two-scale homogenized equation becomes    ∫D∫Y[a(x,y)(curlu0+curlyu1)⋅(curlv0+curlyv1)+b(x,y)(u0+∇yu1)⋅(v0+∇yv1)]dydx  =∫Df(x)⋅v0(x)dx. We first let $$v_0=0$$, $$v_1=0$$ and deduce that   ∫D∫Yb(x,y)(u0+∇yu1)⋅∇yv1dydx=0. For each $$r=1,2,3$$, let $$w^r(x,\cdot)\in L^2(D, H^1_\#(Y)/\mathbb{R})$$ be the solution of the problem   ∫D∫Yb(x,y)(er+∇ywr)⋅∇yψdydx=0  ∀ψ∈L2(D,H#1(Y)/R), (4.1) where $$e_r$$ is the vector in $$\mathbb{R}^3$$ with all the components being 0, except the $$r$$th component, which equals 1. This is the standard cell problem in elliptic homogenization. From this we have   u1(x,y)=wr(x,y)u0r(x). (4.2) Therefore,   ∫D∫Yb(x,y)(u0+∇yu1)⋅v0dxdy=∫Db0(x)u0(x)⋅v0(x)dx, where the positive-definite matrix $$b^0(x)$$ is defined as   bij0(x)=∫Yb(x,y)(ej+∇wj(x,y))⋅(ei+∇ywi(x,y))dy, (4.3) which is the usual homogenized coefficient for elliptic problems with the two-scale coefficient matrix $$b^\varepsilon$$. Let $$v_0=0$$ and $${\frak v_1}=0$$. We have   ∫D∫Ya(x,y)(curlu0+curlyu1)⋅curlyv1dydx=0 for all $$v_1\in L^2(D, \tilde H_\#({\rm curl\,},Y))$$. For each $$r=1,2,3$$, let $$N^r\in L^2(D,\tilde H_\#({\rm curl\,},Y))$$ be the solution of   ∫D∫Ya(x,y)(er+curlyNr)⋅curlyvdydx=0 (4.4) for all $$v\in L^2(D, \tilde H_\#({\rm curl\,},Y))$$. We have   u1=(curlu0(x))rNr(x,y). (4.5) The homogenized coefficient $$a^0$$ is determined by   aij0(x)=∫Ya(x,y)ip(ejp+(curlyNj)p)dy=∫Ya(x,y)(ej+curlyNj)⋅(ei+curlyNi)dy. (4.6) We have   ∫D∫Ya(x,y)(curlu0+curlyu1)⋅curlv0dxdy=∫Da0(x)curlu0(x)⋅curlv0(x)dx. The homogenized problem is   ∫D[a0(x)curlu0(x)⋅curlv0(x)+b0(x)u0(x)⋅v0(x)]dx=∫Df(x)⋅v0(x)dx  ∀v0∈H0(curl,D). (4.7) Following the procedure for deriving the homogenization error (Bensoussan et al., 1978; Jikov et al., 1994), we have the following homogenization error estimate. Theorem 4.1 Assume that $$a\in C(\bar D, C(\bar Y))^{3\times 3}$$, $$u_0\in H^1({\rm curl\,};D)$$, $$N^r\in C^1(\bar D,C(\bar Y))^3$$, $${\rm curl}_yN^r\in C^1(\bar D,C(\bar Y))^3$$, $$w^r\in C^1(\bar D,C^1(\bar Y))$$ for all $$r=1,2,3$$, then2  ‖uε−[u0+∇yu1(⋅,⋅ε)]‖L2(D)3≤cε1/2 and   ‖curluε−[curlu0+curlyu1(⋅,⋅ε)]‖L2(D)3≤cε1/2. The proof of this theorem uses the functions $$G_r$$ and $$g_r$$ defined in (A.2) and (A.3) below. For $$u_0\in H^s({\rm curl\,},D)$$ when $$0<s<1$$, we have the following homogenization error estimate. Theorem 4.2 Assume that $$a\in C(\bar D,C(\bar Y))^{3\times 3}$$, $$u_0\in H^s({\rm curl\,},D)$$, $$N^r\in C^1(\bar D, C(\bar Y))^3$$, $${\rm curl}_yN^r\in C^1(\bar D,C(\bar Y))^3$$ and $$w^r\in C^1(\bar D,C^1(\bar Y))$$ for all $$r=1,2,3$$, then   ‖uε−[u0+∇yu1(⋅,⋅ε)‖L2(D)3≤cεs/(1+s) and   ‖curluε−[curlu0+curlyu1(⋅,⋅ε)]‖L2(D)3≤cεs/(1+s). We present the proof of this theorem in Appendix A. To employ the FE solutions to construct numerical correctors for $$u^{\varepsilon}$$, we define the following operator:   Uε(Φ)(x)=∫YΦ(ε[xε]+εz,{xε})dz. (4.8) Let $$D^\varepsilon$$ be a $$2\varepsilon$$ neighbourhood of $$D$$. Regarding $${\it{\Phi}}$$ as zero when $$x$$ is outside $$D$$, we have   ∫DεUε(Φ)(x)dx=∫D∫YΦ(x,y)dxdy. (4.9) The proof of (4.9) may be found in Cioranescu et al. (2008). We have the following result. Lemma 4.3 Assume that for $$r=1,2,3$$, $${\rm curl}_yN^r(x,y)\in C^1(\bar D, C(\bar Y))^3$$ and $$u_0\in H^s({\rm curl\,},D)$$, then   ‖curlyu1(⋅,⋅ε)−Uε(curlyu1)‖L2(D)3≤cεs. We prove this lemma in Appendix B. We then have the following result. Theorem 4.4 Assume that $$a\in C(\bar D,C(\bar Y))^{3\times 3}$$, $$u_0\in H^s({\rm curl\,},D)$$, $$N^r\in C^1(\bar D,C(\bar Y))^3$$, $${\rm curl}_yN^r\in C^1(\bar D,C(\bar Y))^3$$ and $$w^r\in C^1(\bar D,C^1(\bar Y))$$ for all $$r=1,2,3$$. Then for the full tensor product FE solution $$(u_0^L,u_1^L,{\frak u_1}^L)$$ we have   ‖uε−u0L−Uε(∇uu1L)‖L2(D)3≤c(εs/(1+s)+hLs) and   ‖curluε−curlu0L−Uε(curlyu1L)‖L2(D)3≤c(εs/(1+s)+hLs). Proof. From Lemma 4.3, we have    ‖curluε−curlu0L−Uε(curlyu1L)‖L2(D)3  ≤‖curluε−curlu0−curlyu1(⋅,⋅ε)‖L2(D)3  +‖curlu0−curlu0L‖L2(D)3+‖curlyu1(⋅,⋅ε)−Uε(curlyu1)‖L2(D)3  +‖Uε(curlyu1)−Uε(curlyu1L)‖L2(D)3. Using the fact that $$(\mathcal{U}^\varepsilon({\it{\Phi}}))^2\le \mathcal{U}({\it{\Phi}}^2)$$ and (4.9), we have   ‖Uε(curlyu1)−Uε(curlyu1L)‖L2(D)3≤‖curlyu1−curlyu1L‖L2(D×Y)3≤chLs. This together with (3.2), Theorem 4.2 and Lemma 4.3 gives   ‖curluε−curlu0L−Uε(curlyu1L)‖L2(D)3≤c(εs/(1+s)+hLs). Similarly, we have   ‖uε−u0L−Uε(∇uu1L)‖L2(D)3≤c(εs/(1+s)+hLs). □ For the sparse tensor product FE approximation, we have the following result. Theorem 4.5 Assume that $$a\in C(\bar D,C(\bar Y))^{3\times 3}$$, $$u_0\in H^s({\rm curl\,},D)$$, $$N^r\in C^1(\bar D,C(\bar Y))^3$$, $${\rm curl}_yN^r\in C^1(\bar D,C(\bar Y))^3$$ and $$w^r\in C^1(\bar D,C^1(\bar Y))$$ for all $$r=1,2,3$$. Then for the sparse tensor product FE solution $$(\hat u_0^L,\hat u_1^L,\hat{\frak u}_1^L)$$ we have   ‖uε−u^0L−Uε(∇yu^1L)‖L2(D)3≤c(εs/(1+s)+Ln/2hLs) and   ‖curluε−curlu^0L−Uε(curlyu^1L)‖L2(D)3≤c(εs/(1+s)+Ln/2hLs). 4.2 Multiscale problems For multiscale problems, we do not have an explicit homogenization rate of convergence. However, for the case where $$\varepsilon_i/\varepsilon_{i+1}$$ is an integer for all $$i=1,\ldots,n-1$$ we can derive a corrector for the solution $$u^{\varepsilon}$$ of the multiscale problem from the FE solutions of the multiscale homogenized problem. For each function $$\phi\in L^1(D),$$ which is understood as 0 outside $$D$$, we define a function in $$L^1(D\times{\bf Y})$$:   Tnε(ϕ)(x,y)=ϕ(ε1[xε1]+ε2[y1ε2/ε1]+⋯+εn[yn−1εn/εn−1]+εnyn). Letting $$D^{\varepsilon_1}$$ be the $$2\varepsilon_1$$ neighbourhood of $$D$$, we have   ∫Dϕdx=∫Dε1∫Y1⋯∫YnTnε(ϕ)dyn⋯dy1dx (4.10) for all $$\phi \in L^1(D)$$. If a sequence $$\{\phi^\varepsilon\}_\varepsilon$$$$(n+1)$$-scale converges to $$\phi(x,y_1,\ldots,y_n)$$ then   Tnε(ϕ)⇀ϕ(x,y1,…,yn) in $$L^2(D\times Y_1\times\ldots\times Y_n)$$. Thus, when $$\varepsilon\to 0$$,   Tnε(curluε)⇀curlu0+curly1u1+⋯+curlynun (4.11) and   Tnε(uε)⇀u0+∇y1u1+⋯+∇ynun (4.12) in $$L^2(D\times{\bf Y})^3$$. To deduce an approximation of $$u^{\varepsilon}$$ in $$H({\rm curl\,},D)$$ in terms of the FE solution, we use the operator $$\mathcal{U}_n^\varepsilon$$ , which is defined as   Unε(Φ)(x) =∫Y1⋯∫YnΦ(ε1[xε1]+ε1t1,ε2ε1[ε1ε2{xε1}]   +ε2ε1t2,⋯, εnεn−1[εn−1εn{xεn−1}]+εnεn−1tn,{xεn})dtn⋯dt1 for all functions $${\it{\Phi}}\in L^1(D\times{\bf Y})$$. For each function $${\it{\Phi}}\in L^1(D\times{\bf Y})$$ we have   ∫Dε1Unε(Φ)dx=∫D∫YΦ(x,y)dydx. (4.13) The proofs for these facts may be found in Cioranescu et al. (2008). We then have the following corrector result. Proposition 4.6 The solution $$u^{\varepsilon}$$ of problem (2.5) and the solution $$(u_0,\{u_i\}\,\{{\frak u_i}\})$$ of problem (2.10) satisfies   limε→0‖uε−[u0+Unε(∇y1u1)+⋯+Unε(∇ynun)‖L2(D)3=0 (4.14) and   limε→0‖curluε−[curlu0+Unε(curly1u1)+⋯+Unε(curlynun)]‖L2(D)3=0. (4.15) Proof. We consider the expression    ∫D∫Y[Tnε(aε)(Tnε(curluε)−(curlu0+curly1u1+⋯+curlynun)) ⋅(Tnε(curluε)−(curlu0+curly1u1+⋯+curlynun)) +Tnε(bε)(Tnε(uε)−(u0+∇y1u1+⋯+∇ynun))⋅(Tnε(uε)−(u0+∇y1u1+⋯+∇ynun))dydx. Using (2.5), (2.10), (4.10), (4.11) and (4.12), we deduce that this expression converges to 0. From (2.1) we have   limε→0‖Tnε(curluε)−(curlu0+curly1u1+⋯+curlynun)‖L2(D×Y1×…×Yn)3=0 and   limε→0‖Tnε(uε)−(u0+∇y1u1+⋯+∇ynun)‖L2(D×Y1×⋯×Yn)3=0. From (4.13) and the fact that $$\mathcal{U}^\varepsilon_n({\it{\Phi}})^2\le \mathcal{U}^\varepsilon_n({\it{\Phi}}^2)$$, we have    ∫D|Unε(Tnε(curluε)−(curlu0+curly1u1+⋯+curlynun)(x)|2dx ≤∫DUnε(|Tnε(curluε)−(curlu0+curly1u1+⋯+curlynun|2)(x)|dx ≤∫D∫Y|Tnε(curluε)−(curlu0+curly1u1+⋯+curlynun|2)dydx, which converges to 0 when $$\varepsilon\to 0$$. Using $$\mathcal{U}^\varepsilon_n(\mathcal{T}^\varepsilon_n({\it{\Phi}}))={\it{\Phi}}$$, we get (4.15). We derive (4.14) similarly. □ We then deduce the numerical corrector result. Theorem 4.7 For the full tensor product FE approximation solution $$\boldsymbol{u}^L=(u_0^L,\{u_i^L\},\{{\frak u_i}^L\})$$ in (3.1), we have   limε→0L→∞‖uε−[u0L+Unε(∇y1u1L)+⋯+Unε(∇ynunL)]‖L2(D)3=0 (4.16) and   limε→0L→∞‖curluε−[curlu0L+Unε(curly1u1L)+⋯+Unε(curlynunL)]‖L2(D)3=0. (4.17) Proof. We note that    ‖Unε(curly1u1+⋯+curlynun)−Unε(curly1u1L+⋯+curlynunL)‖L2(D)3 ≤∫DUnε(|(curly1u1+⋯+curlynun)−(curly1u1L+⋯+curlynunL)|2)(x)dx ≤∫D∫Y|(curly1u1+⋯+curlynun)−(curly1u1L+⋯+curlynunL)|2dydx, which converges to 0 when $$L\to \infty$$. From this and (4.15), we get (4.17). We obtain (4.16) in the same way. □ Remark 4.8 As $${\bf V}^{\lceil L/n\rceil}\subset\hat{\bf V}^L$$, the result in Theorem 4.7 also holds for the sparse tensor product FE solution $$\widehat {\bf{u}}^L$$. Since we do not have an explicit homogenization error for problems with more than two scales, we do not distinguish the two cases of full and sparse tensor FE approximations. 5. Regularity of $$\boldsymbol{N^r}$$, $$\boldsymbol{w^r}$$ and $$\boldsymbol{u_0}$$ We show in this section that the regularity requirements for obtaining the sparse tensor product FE error estimate and the homogenization error estimate in the previous sections are achievable. We present the results for the two-scale case in detail. The multiscale case is similar; we summarize it in Remark 5.7. We first prove the following lemma. Lemma 5.1 Let $$\psi\in H_\#({\rm curl\,},Y)\bigcap H_\#({\rm div},Y)$$. Assume further that $$\int_Y\psi(y)\,{\rm d}y=0$$. Then $$\psi\in H^1_\#(Y)^3$$ and   ‖ψ‖H1(Y)3≤c(‖curlyψ‖L2(Y)3+‖divyψ‖L2(Y)). Proof. Let $$\omega\subset\mathbb{R}^3$$ be a smooth domain such that $$\omega\supset Y$$. Let $$\eta\in \mathcal{D}(\omega)$$ be such that $$\eta(y)=1$$ when $$y\in Y$$. We have   curly(ηψ)=ηcurlyψ+∇yη×ψ∈L2(ω)3 and   divy(ηψ)=∇yη⋅ψ+ηdivyψ∈L2(ω)3. Together with the zero boundary condition, we conclude that $$\eta\psi\in H^1(\omega)^3$$ so $$\psi\in H^1(Y)^3$$. We note that   ∫Y(divyψ(y)2+|curlyψ(y)|2)dy=∑i,j=13∫Y(∂ψi∂yj)2+∑i≠j∫Y∂ψi∂yi∂ψj∂yjdy−∑i≠j∫Y∂ψj∂yi∂ψi∂yjdy. Assume that $$\psi$$ is a smooth periodic function. We have   ∫Y∂ψi∂yi∂ψj∂yjdy=∫Y[∂∂yi(ψi∂ψj∂yj)−ψi∂2ψj∂yi∂yj]dy=−∫Yψi∂2ψj∂yi∂yjdy as $$\psi$$ is periodic. Similarly, we have   ∫Y∂ψi∂yj∂ψj∂yidy=∫Y[∂∂yj(ψi∂ψj∂yi)−ψi∂2ψj∂yj∂yi]dy=−∫Yψi∂2ψj∂yj∂yidy. Thus,   ∫Y∂ψi∂yi∂ψj∂yjdy=∫Y∂ψi∂yj∂ψj∂yidy. Therefore,   ‖∇yψ‖L2(Y)32=‖divyψ‖L2(Y)2+‖curlyψ‖L2(Y)32. Using a density argument, this holds for all $$\psi\in H^1_\#(Y)^3$$. As $$\int_Y\psi(y)\,{\rm d}y=0$$, from the Poincaré inequality we deduce   ‖ψ‖H1(Y)3≤c(‖divyψ‖L2(Y)+‖curlyψ‖L2(Y)3). □ Lemma 5.2 Let $$\alpha\in C^1_\#(\bar Y)^{3\times 3}$$ be uniformly bounded, positive definite and symmetric for all $$y\in \bar Y$$. Let $$F\in L^2(Y)$$, extending periodically to $$\mathbb{R}^3$$. Let $$\psi\in H^1_\#(Y)^3$$ satisfy the equation   curly(α(y)curlyψ(y))=F(y). Then $${\rm curl}_y\psi\in H^1_\#(Y)^3$$ and   ‖curlyψ‖H1(Y)3≤c(‖F‖L2(Y)3+‖ψ‖H1(Y)3). Proof. Let $$\omega\supset Y$$ be a smooth domain. Let $$\eta\in \mathcal{D}(\omega)$$ be such that $$\eta(y)=1$$ for $$y\in Y$$. We have   curly(αcurly(ηψ)) =curly(αηcurlyψ)+curly(α∇yη×ψ) =ηcurly(αcurlyψ)+∇yη×(αcurlyψ)+curly(α∇yη×ψ). Let $$U=\alpha{\rm curl}_y(\eta\psi)$$. We have   ‖curlyU‖L2(ω)3≤c(‖F‖L2(ω)+‖ψ‖H1(ω)3)≤c(‖F‖L2(Y)3+‖ψ‖H1(Y)3). Further,   ‖U‖L2(ω)3=‖α(∇yη×ψ+ηcurlyψ)‖|L2(ω)3≤c‖ψ‖H1(ω)3≤c‖ψ‖H1(Y)3. As $$\eta\in\mathcal{D}(\omega)$$, $$U$$ has compact support in $$\omega$$ so $$U$$ belongs to $$H_0({\rm curl\,},\omega)$$. Thus, we can write   U=z+∇Φ, where $$z\in H^1_0(\omega)^3$$ and $${\it{\Phi}}\in H^1_0(\omega)$$ satisfy   ‖z‖H1(ω)3≤c‖U‖H(curl,ω)  and  ‖Φ‖H1(ω)≤c‖U‖H(curl,ω). From $${\rm div}_y(\alpha^{-1}U)=0$$ we deduce that   divy(α−1∇Φ)=−divy(α−1z)∈L2(ω). Since $$\alpha\in C^1(\bar\omega)^{3\times 3}$$ and is uniformly bounded and positive definite, $$\alpha^{-1}\in C^1(\bar\omega)^{3\times 3}$$ and is uniformly positive definite. Therefore, $${\it{\Phi}}\in H^2(\omega)$$ and satisfies   ‖Φ‖H2(ω)≤c‖z‖H1(ω)3≤c‖U‖H(curl,ω). Thus, $$U\in H^1(\omega)^3$$ and $$\|U\|_{H^1(\omega)^3}\le c\|U\|_{H({\rm curl\,},\omega)}\le c(\|F\|_{L^2(Y)^3}+\|\psi\|_{H^1(Y)^3})$$. From $${\rm curl}_y(\eta\psi)=\alpha^{-1}U$$, we deduce that $${\rm curl}_y(\eta\psi)\in H^1(\omega)^3$$ so $${\rm curl}_y\psi\in H^1(Y)^3$$ and   ‖curlyψ‖H1(Y)3≤c(‖F‖L2(Y)3+‖ψ‖H1(Y)3). □ We then prove the following result on the regularity of $$N^r$$. Proposition 5.3 Assume that $$a(x,y)\in C^1(\bar D,C^2(\bar Y))^{3\times 3}$$; then $${\rm curl}_yN^r(x,y)\in C^1(\bar D,C(\bar Y))^3$$ and we can choose a version of $$N^r$$ in $$L^2(D,\tilde H_\#({\rm curl\,},Y))$$ so that $$N^r(x,y)\in C^1(\bar D, C(\bar Y))^3$$. Proof. We can choose a version of $$N^r$$ so that $${\rm div}_yN^r=0$$. Indeed, let $${\it{\Phi}}(x,\cdot)\in L^2(D, H^1_\#(Y))$$ be such that $$\Delta_y{\it{\Phi}}=-{\rm div}_yN^r$$; then $${\rm curl}_y(N^r+\nabla_y{\it{\Phi}})={\rm curl}_y N^r$$ and $${\rm div}_y(N^r+\nabla_y{\it{\Phi}})=0$$. Further we can choose $$N^r$$ so that $$\int_YN^r(x,y)\,{\rm d}y=0$$. From Lemma 5.1, we have   ‖Nr(x,⋅)‖H1(Y)3≤c‖curlyNr(x,⋅)‖L2(Y)3, which is uniformly bounded with respect to $$x$$. From (4.4) and Lemma 5.2, we deduce that   ‖curlyNr(x,⋅)‖H1(Y)3≤c‖curly(a(x,⋅)er‖L2(Y)3+‖Nr(x,⋅)‖H1(Y)3, which is uniformly bounded with respect to $$x$$. For each index $$q=1,2,3$$, we have that $${\rm curl}_y{\partial\over\partial y_q}N^r(x,\cdot)$$ is uniformly bounded in $$L^2(Y)$$ and $${\rm div}_y{\partial\over\partial y_q}N^r(x,\cdot)=0$$. Therefore, from Lemma 5.1, $${\partial\over\partial y_q}N^r(x,\cdot)$$ is uniformly bounded in $$H^1(Y)$$. We note that   ∂∂yq(curly(a(x,⋅)curlyNr))=−∂∂yqcurly(a(x,⋅)er)∈L2(Y). Thus,   curly(acurly∂Nr∂yq)=∂∂yq(curly(a(x,y)curlyNr))−curly(∂a∂yqcurlyNr)∈L2(Y). From Lemma 5.2 we deduce that $${\rm curl}_y{\partial N^r\over\partial y_q}(x,\cdot)$$ is uniformly bounded in $$H^1(Y)$$ so that $${\rm curl}_y N^r(x,\cdot)$$ is uniformly bounded in $$H^2(Y)\subset C(\bar Y)$$. We now show that $${\rm curl}_yN^r\in C^1(\bar D,H^2(Y))^3\subset C^1(\bar D,C(\bar Y))^3$$. Fix $$h\in\mathbb{R}^3$$. From (4.4) we have   curly(a(x,y)curly(Nr(x+h,y)−Nr(x,y))) =−curly((a(x+h,y)−a(x,y))er) −curly((a(x+h,y)−a(x,y))curlyNr(x+h,y)). The smoothness of $$a$$ and the uniform boundedness of $${\rm curl}_yN^r(x,\cdot)$$ in $$L^2(Y)^3$$ gives   limh→0‖curly(Nr(x+h,⋅)−Nr(x,⋅))‖L2(Y)3=0. (5.1) From Lemma 5.1 we have that $$N^r(x+h,\cdot)-N^r(x,\cdot)\in H^1(Y)^3$$ and   ‖Nr(x+h,⋅)−Nr(x,⋅)‖H1(Y)3≤c‖curly(Nr(x+h,⋅)−Nr(x,⋅))‖L2(Y)3, which converges to 0 when $$|h|\to 0$$. From Lemma 5.2, we have    ‖curly(Nr(x+h,⋅)−Nr(x,⋅))‖H1(Y)3 ≤‖−curly((a(x+h,⋅)−a(x,⋅))er)−curly((a(x+h,⋅)−a(x,⋅))curlyNr(x+h,⋅))‖L2(Y)3 +‖Nr(x+h,⋅)−Nr(x,⋅)‖H1(Y)3→0  when |h|→0. (5.2) We have further that   curly(a(x,y)curly∂∂yq(Nr(x+h,y)−Nr(x,y)))=−curly(∂a∂yq(x,y)curly(Nr(x+h,y)−Nr(x,y)))  −∂∂yqcurly((a(x+h,y)−a(x,y))er)−∂∂yqcurly((a(x+h,y)−a(x,y))curlyNr(x+h,y)). (5.3) From this we have   ‖curly∂∂yq(Nr(x+h,y)−Nr(x,y))‖L2(Y)3 ≤c‖curly(Nr(x+h,⋅)−Nr(x,⋅))‖L2(Y)3  +c‖a(x+h,⋅)−a(x,⋅)‖W1,∞(Y)3→0  when |h|→0, so from Lemma 5.1 we have   ‖∂∂yq(Nr(x+h,y)−Nr(x,y))‖H1(Y)3→0  when |h|→0. As the right-hand side of (5.3) converges to 0 in the $$L^2(Y)^3$$ norm when $$|h|\to 0$$, we deduce from Lemma 5.2 that   ‖curly∂∂yq(Nr(x+h,y)−Nr(x,y))‖H1(Y)3→0  when |h|→0. (5.4) We have   curly[a(x,y)curly(Nr(x+h,y)−Nr(x,y)h)] =−curly((a(x+h,y)−a(x,y)h)er)  −curly(a(x+h,y)−a(x,y)hcurlyNr(x+h,y)). Let $$\chi^r(x,\cdot)\in \tilde H_\#({\rm curl\,},Y)$$ with $${\rm div}_y\chi^r(x,)=0$$ be the solution of the problem   curly(a(x,y)curlyχr(x,⋅))=−curly(∂a∂xqer)−curly(∂a∂xqcurlyNr(x,y)). We deduce that   curly(a(x,y)curly(Nr(x+h,y)−Nr(x,y)h−χr(x,y))) =−curly((a(x+h,y)−a(x,y)h−∂a∂xq(x,y))er) −curly((a(x+h,y)−a(x,y)h−∂a∂xq(x,y))curlyNr(x+h,y)) −curly(∂a∂xq(x,y)curly(Nr(x+h,y)−Nr(x,y))):=I1. (5.5) Let $$h\in \mathbb{R}^3$$ be a vector with all components 0 except for the $$q$$th component. We have   ‖curly(Nr(x+h,⋅)−Nr(x,⋅)h−χr(x,⋅))‖L2(Y)3 ≤c‖a(x+h,⋅)−a(x,⋅)h−∂a∂xq(x,⋅)‖L∞(Y) +c‖curly(Nr(x+h,⋅)−Nr(x,⋅)‖L2(Y)3, (5.6) which converges to 0 when $$|h|\to 0$$ due to (5.1). Thus, we deduce from Lemma 5.1 that   lim|h|→0‖Nr(x+h,⋅)−Nr(x,⋅)h−χr(x,⋅)‖H1(Y)3=0. (5.7) From Lemma 5.2, we have    lim|h|→0‖curly(Nr(x+h,⋅)−Nr(x,⋅)h−χr(x,⋅))‖H1(Y)3 ≤lim|h|→0‖I1(x,⋅)‖L2(Y)3+‖Nr(x+h,⋅)−Nr(x,⋅)h−χr(x,⋅)‖H1(Y)3=0 (5.8) due to (5.2) and (5.7). Let $$p=1,2,3$$. We then have   curly(a(x,y)curly∂∂yp(Nr(x+h,y)−Nr(x,y)h−χr(x,y))) =−curly(∂a∂yp(x,y)curly(Nr(x+h,y)−Nr(x,y)h−χr(x,y))) −∂∂ypcurly((a(x+h,y)−a(x,y)h−∂a(x,y)∂xq)er) −∂∂ypcurly((a(x+h,y)−a(x,y)h−∂a∂xq(x,y))curlyNr(x+h,y)) −∂∂ypcurly(∂a∂xq(x,y)curly(Nr(x+h,y)−Nr(x,y))), which converges to 0 in $$L^2(Y)$$ for each $$x$$ due to (5.4), (5.8) and the uniform boundedness of $$\|{\rm curl\,} N^r(x,\cdot)\|_{H^2(Y)^3}$$. We have   ‖curly∂∂yp(Nr(x+h,⋅)−Nr(x,⋅)h−χr(x,⋅))‖L2(Y)3  ≤c‖curly(Nr(x+h,⋅)−Nr(x,⋅)h−χr(x,⋅))‖L2(Y)3  +c‖a(x+h,⋅)−a(x,⋅)h−∂a(x,⋅)∂xq‖W1,∞(Y)3+c‖curly(Nr(x+h,⋅)−Nr(x,⋅)‖H1(Y)3, which converges to 0 when $$|h|\to 0$$, so from Lemma 5.1,   lim|h|→0‖∂∂yp(Nr(x+h,⋅)−Nr(x,⋅)h−χr(x,⋅))‖H1(Y)3=0. (5.9) Therefore, $$N^r\in C^1(\bar D,H^2(Y))^3\subset C^1(\bar D,C(\bar Y))^3$$. We then get from Lemma 5.2 that   lim|h|→0‖curly∂∂yp(Nr(x+h,⋅)−Nr(x,⋅)h−χr(x,⋅))‖H1(Y)3=0. Thus, $${\rm curl}_yN^r\in C^1(\bar D,H^2(Y))^3\subset C^1(\bar D,C(\bar Y))^3$$. □ Proposition 5.4 Assume that $$b(x,y)\in C^1(\bar D,C^2(\bar Y))^{3\times 3}$$. The solution $$w^r$$ of cell problem (4.1) belongs to $$C^1(\bar D, C^1(\bar Y))$$. Proof. The cell problem (4.1) can be written as   −∇y⋅(b(x,y)∇ywr(x,y))=∇y(b(x,y)er). Fixing $$x\in \bar D$$, the right-hand side is bounded uniformly in $$H^1(Y)$$ so $$w^r(x,\cdot)$$ is uniformly bounded in $$H^3(Y)$$ from elliptic regularity (see McLean, 2000, Theorem 4.16). For $$h\in \mathbb{R}^3$$, we note that   −∇y⋅[b(x,y)∇y(wr(x+h,y)−wr(x,y))] =∇y⋅[(b(x+h,y)−b(x,y))er] +∇y⋅[(b(x+h,y)−b(x,y))∇ywr(x+h,y)]:=i1. As $$\int_Yw^r(x,y)\,{\rm d}y=0$$, we have   ‖wr(x+h,⋅)−wr(x,⋅)‖H1(Y) ≤c‖∇y(wr(x+h,⋅)−wr(x,⋅))‖L2(Y) ≤c‖(b(x+h,⋅)−b(x,⋅))er‖L2(Y) +c‖(b(x+h,⋅)−b(x,⋅))∇ywr(x+h,⋅)‖L2(Y), which converges to 0 when $$|h|\to 0$$. Fixing $$x\in\bar D$$, we then have from McLean (2000, Theorem 4.16) that   ‖wr(x+h,⋅)−wr(x,⋅)‖H3(Y)≤‖wr(x+h,⋅)−wr(x,⋅)‖H1(Y)+‖i1(x,⋅)‖H1(Y), (5.10) which converges to 0 when $$|h|\to 0$$. Fixing an index $$q=1,2,3$$, let $$h\in \mathbb{R}^3$$ be a vector whose components are all zero except the $$q$$th component. Let $$\eta(x,\cdot)\in H^1_\#(Y)/\mathbb{R}$$ be the solution of the problem   −∇y⋅[b(x,y)∇yη(x,y)]=∇y⋅[∂b∂xqer]+∇y⋅[∂b∂xq∇ywr(x,y)]. We have    −∇y⋅[b(x,y)∇y(wr(x+h,y)−wr(x,y)h−η(x,y))]  =∇y⋅[(b(x+h,y)−b(x,y)h−∂b∂xq(x,y))er]  +∇y⋅[(b(x+h,y)−b(x,y)h−∂b∂xq(x,y))∇ywr(x+h,y)]  +∇y⋅[∂b(x,y)∂xq(∇ywr(x+h,y)−∇ywr(x,y))]:=i2. From (5.10) and the regularity of $$b$$, $$\lim_{|h|\to 0}\|i_2(x,\cdot)\|_{H^1(Y)}=0$$. As $$\int_Yw^r(x,y)\,{\rm d}y=0$$ and $$\int_Y\eta(x,y)\,{\rm d}y=0$$, we have   lim|h|→0‖wr(x+h,⋅)−wr(x,⋅)h−η(x,⋅)‖H1(Y)=0. Therefore from McLean (2000, Theorem 4.16), we have   ‖wr(x+h,⋅)−wr(x,⋅)h−η(x,⋅)‖H3(Y)≤‖wr(x+h,⋅)−wr(x,⋅)h−η(x,⋅)‖H1(Y)+‖i2(x,⋅)‖H1(Y), which converges to 0 when $$|h|\to 0$$. Thus, $$w^r\in C^1(\bar D,H^3(Y))\subset C^1(\bar D,C^1(\bar Y))$$. □ For the regularity of the solution $$u_0$$ of the homogenized problem (4.7) we have the following result. Proposition 5.5 Assume that $$D$$ is a Lipschitz polygonal domain, and the coefficient $$a(\cdot,y)$$, as a function of $$x$$, is Lipschitz, uniformly with respect to $$y$$; then there is a constant $$0<s<1$$ so that $${\rm curl\,} u_0\in H^s(D)$$. Proof. When $$a(x,y)$$ is Lipschitz with respect to $$x$$, from (4.4), $$\|{\rm curl}_y N^r(x,\cdot)\|_{L^2(Y)}$$ is a Lipschitz function of $$x$$, so from (4.6) we have that $$a^0$$ is Lipschitz with respect to $$x$$. As $$a^0$$ is positive definite, $$(a^0)^{-1}$$ is Lipschitz. Let $$U=a^0{\rm curl\,} u_0$$. We have from (4.7) that $$U\in H({\rm curl\,},D)$$, $${\rm div}((a^0)^{-1}U)=0$$ and $$(a^0)^{-1}U\cdot \nu=0$$ on $$\partial D,$$ where $$\nu$$ is the outward normal vector on $$\partial D$$. The conclusion follows from Hiptmair (2002, Lemma 4.2). □ Remark 5.6 If $$a^0$$ is isotropic, we have from (4.7) that   curlcurlu0=−(a0)−1∇a0×curlu0−(a0)−1b0u0+(a0)−1f∈L2(D)3 so $$u_0\in H^1({\rm curl},D)$$. However, even if $$a$$ is isotropic, $$a^0$$ may not be isotropic. Remark 5.7 The homogenized equation for the multiscale case is determined as follows. We denote by $$a^n(x,\boldsymbol{y})=a(x,\boldsymbol{y})$$. Recursively, for $$i=1,\ldots,n-1$$, the $$i$$-th level homogenized coefficient is determined as follows. For $$r=1,2,3$$, let $$N_{i+1}^r\in L^2(D\times{\bf Y}_i,\tilde H({\rm curl}_{y_{i+1}},Y_{i+1}))$$ be the solution of the cell problem   ∫D∫Y1…∫Yi+1ai+1(x,yi,yi+1)(er+curlyi+1Ni+1r)⋅curlyi+1ψdyi+1dyidx=0 for all $$\psi\in L^2(D\times{\bf Y}_i,\tilde H({\rm curl}_{y_{i+1}},Y_{i+1}))$$. The $$i$$-th level homogenized coefficient $$a^{i}$$ is determined by   arsi(x,yi)=∫Yi+1ai+1(x,yi,yi+1)(es+curlyi+1Ni+1s)⋅(er+curlyi+1Ni+1r)dyi+1. Let $$b^n(x,\boldsymbol{y})=b(x,\boldsymbol{y})$$. Similarly, let $$w_{i+1}^{r}\in L^2(D\times{\bf Y}_i,H^1_\#(Y_{i+1})/\mathbb{R})$$ be the solution of the problem   ∫D∫Y1…∫Yi+1bi+1(x,yi,yi+1)(er+∇yi+1wi+1r)⋅∇yi+1ψdyi+1dyidx=0 for all $$\psi\in L^2(D\times{\bf Y}_i,H^1_\#(Y_{i+1})/\mathbb{R})$$. The $$i$$-th level homogenized coefficient $$b^i$$ is determined by   brsi(x,yi)=∫Yi+1bi+1(x,yi,yi+1)(es+∇yi+1wi+1s)⋅(er+∇yi+1wi+1r)dyi+1. We then have the equation    ∫D∫Yi[ai(x,yi)(curlu0+curly1u1+⋯+curlyiui)⋅(curlv0+curly1v1+⋯+curlyivi)dyidx  +bi(x,yi)(u0+∇y1u1+⋯+∇yiui)⋅(v0+∇y1v1+⋯+∇yivi)]dyidx=∫Df(x)⋅v0(x)dx. The coefficients $$a^0(x)$$ and $$b^0(x)$$ are the homogenized coefficients. We have   ui(x,yi) = [curlu0(x)r+curly1u1(x,y1)r+⋯+curlyi−1ui−1(x,yi−1)r]Nir(x,yi) =curlu0(x)r0(δr0r1+curly1N1r0(x,y1)r1)(δr1r2+curly2N2r1(x,y2)r2)⋯ (δri−2ri−1+curlyi−1Ni−1ri−2(x,yi−1)ri−1)Niri−1(x,yi). If $$a(x,\boldsymbol{y})\in C^1(\bar D,C^2(\bar Y_1,\ldots,C^2(\bar Y_n),\ldots))^{3\times 3}$$, by following the same procedure as above, we can show inductively that $${\rm curl}_{y_i}N_i^r(x,\boldsymbol{y}_i)\in C^1(\bar D,C^2(\bar Y_1,\ldots,C^2(\bar Y_{i-1},H^2(Y_i)),\ldots))$$ and $$a^i(x,\boldsymbol{y}_i)\in C^1(\bar D,C^2(\bar Y_1,\ldots,C^2(\bar Y_i),\ldots))$$. Thus, if $$u_0\in H^s({\rm curl\,},D)$$ for $$0<s\le 1$$, $$u_i\in \hat{\mathcal{H}}_i^s$$. Similarly, we can show that if $$b\in C^1(\bar D, C^2(\bar Y_1,\ldots,C^2(\bar Y_n)\ldots))$$ then $$w^{ir}\in C^1(\bar D,C^2(\bar Y_1,\ldots,C^2(\bar Y_{i-1},H^3(Y_i))\ldots))$$. As   ui=u0r0(x)(δr0r1+∂w1r0∂y1r1(x,y1))…(δri−2ri−1+∂wi−1ri−2∂y(i−1)ri−1(x,yi−1))wiri−1(x,yi), if $$u_0\in H^s(D)$$, $${\frak u_i}\in \hat{\frak H}_i^s$$. 6. Numerical results The detail spaces $$\mathcal{V}^l$$ and $$\mathcal{V}^l_\#$$, which are difficult to construct in numerical implementations, are defined via orthogonal projection in Section 3.2. We employ Riesz basis functions and define equivalent norms, which facilitate the construction of these spaces. We make the following assumption. Assumption 6.1 (i) For each multidimensional vector $$j \in \mathbb{N}_0^d$$, there exists a set of indices $$I^j \subset \mathbb{N}^d_0$$ and a set of basis functions $$\phi^{jk}\in L^2(D)$$ for $$k\in I^j$$, such that $$V^l = \text{span}\left\{\phi^{jk} : |\,j|_{\infty}\le l\right\}$$. There are constants $$c_2>c_1>0$$ such that if $$\phi = \sum_{|\,j|_{\infty}\leq l,k\in I^j}\phi^{jk}c_{jk}\in V^l$$, then the following norm equivalences hold:   c1∑|j|∞≤lk∈Ij|cjk|2≤‖ϕ‖L2(D)2≤c2∑|j|∞≤lk∈Ij|cjk|2, where $$c_1$$ and $$c_2$$ are independent of $$\phi$$ and $$l$$. (ii) For the space $$L^2(Y)$$, for each $$j \in \mathbb{N}_0^d$$, there exists a set of indices $$I^j_0 \subset \mathbb{N}_0^d$$ and a set of basis functions $$\phi^{jk}_0\in L^2(Y)$$, $$k\in I^j_0$$, such that $$V^l_\# = \text{span}\{\phi^{jk}_0 : |\,j|_{\infty}\le l\}$$. There are constants $$c_4>c_3>0$$ such that if $$\phi = \sum_{|\,j|_{\infty}\leq l,k\in I^{\,j}_{\,0}}\phi^{jk}_0c_{jk}\in V^l$$ then   c3∑|j|∞≤lk∈I0j|cjk|2≤‖ϕ‖L2(Y)2≤c4∑|j|∞≤lk∈I0j|cjk|2, where $$c_3$$ and $$c_4$$ are independent of $$\phi$$ and $$l$$. Because of the norm equivalence, we can use $$\mathcal{V}^l=\text{span}\{\phi^{jk} : |\,j|_{\infty}= l\}$$ and $$\mathcal{V}^l_\#=\text{span}\{\phi^{jk}_0 : |\,j|_{\infty}= l\}$$ to construct the sparse tensor product FE spaces. Example 6.2 (i) We can construct a hierarchical basis for $$L^2(0,1)$$ as follows. We first take three piecewise linear functions as the basis for level $$j=0$$: $$\psi^{01}$$ obtains values $$(1,0)$$ at $$(0,1/2)$$ and is 0 in $$(1/2,1)$$, $$\psi^{02}$$ is piecewise linear and obtains values $$(0, 1, 0)$$ at $$(0, 1/2, 1)$$ and $$\psi^{03}$$ obtains values $$(0,1)$$ at $$(1/2, 1)$$ and is 0 in $$(0,1/2)$$. The basis functions for other levels are constructed from the wavelet function $$\psi$$ that takes values $$(0,-1,2,-1,0)$$ at $$(0,1/2,1,3/2,2)$$, the left boundary function $$\psi^{\rm left}$$ taking values $$(-2,2,-1,0)$$ at $$(0,1/2,1,3/2)$$ and the right boundary function $$\psi^{\rm right}$$ taking values $$(0, -1,2,-2)$$ at $$(1/2,1,3/2,2)$$. For levels $$j\geq 1$$, $$I^j=\{1,2,\ldots,2^j\}$$. The wavelet basis functions are defined as $$\psi^{j1}(x) = 2^{j/2}\psi^{\rm left}(2^j x)$$, $$\psi^{jk}(x)=2^{j/2}\psi(2^j x - k + 3/2)$$ for $$k = 2, \ldots, 2^j-1$$ and $$\psi^{j2^j} = 2^{j/2}\psi^{\rm right}(2^j x - 2^j+2)$$. This base satisfies Assumption 6.1 (i). (ii) For $$Y = (0,1)$$, we can construct a hierarchy of periodic basis functions for $$L^2(Y)$$ that satisfies Assumption 6.1 (ii) from those in (i). For level 0, we exclude $$\psi^{01}$$, $$\psi^{03}$$ and include the periodic piecewise linear function that takes values $$(1,0,1)$$ at $$(0,1/2,1),$$ respectively. At other levels, the functions $$\psi^{\rm left}$$ and $$\psi^{\rm right}$$ are replaced by the piecewise linear functions that take values $$(0,2, -1, 0)$$ at $$(0,1/2,1,3/2)$$ and values $$(0, -1,2,0)$$ at $$(1/2,1, 3/2 ,2),$$ respectively. When $$D=(0,1)^d$$, the basis functions can be constructed by taking the tensor products of the basis functions in $$(0,1)$$. They satisfy Assumption 6.1 after appropriate scaling (see Griebel & Oswald, 1995). Remark 6.3 When the norm equivalence for the basis functions in $$L^2(D)$$ and in $$L^2(Y)$$ does not hold, in many cases, we can still prove a rate of convergence similar to those in Lemmas 3.5 and 3.6 for the sparse tensor product FE approximations. For example, with the division of the domain $$D$$ into sets of triangles $$\mathcal{T}^l$$, the set of continuous piecewise linear functions with value 1 at one vertex and 0 at all the others forms a basis of $$V^l$$. Let $$S^l$$ be the set of vertices of the set of simplices $$\mathcal{T}^l$$. We can define $$\mathcal{V}^l$$ as the linear span of functions that are 1 at a vertex in $$S^l\setminus S^{l-1}$$ and 0 at all the other vertices. We can then construct the sparse tensor product FE approximations with these spaces but the norm equivalence does not hold. A rate of convergence for sparse tensor product FEs similar to those in Lemmas 3.5 and 3.6 can be deduced (see e.g, Hoang, 2008). In the first example, we consider a two-scale Maxwell-type equation in the two-dimensional domain $$D=(0,1)^2$$. The coefficients   a(x,y)=(1+x1)(1+x2)(1+cos2⁡2πy1)(1+cos2⁡2πy2) and   b(x,y)=1(1+x1)(1+x2)(1+cos2⁡2πy1)(1+cos2⁡2πy2). We can compute the homogenized coefficients exactly. In this case,   a0=4(1+x1)(1+x2)9  and  b0=23(1+x1)(1+x2). We choose   f=(49(1+x1)(1+2x2−x1)+23(1+x1)(1+x2)x1x2(1−x2)49(1+x2)(1+2x1−x2)+23(1+x1)(1+x2)x1x2(1−x1)) so that the solution to the homogenized equation is   u0=(x1x2(1−x2)x1x2(1−x1)). In Fig. 1, we plot the energy error versus the mesh size for the sparse tensor product FE approximations of the two-scale homogenized Maxwell-type problem. The figure agrees with the error estimate in Proposition 3.8. Fig. 1. View largeDownload slide The sparse tensor energy error $$B(\boldsymbol{u}-\widehat {\bf{u}}^L,\boldsymbol{u}-\widehat {\bf{u}}^L).$$ Fig. 1. View largeDownload slide The sparse tensor energy error $$B(\boldsymbol{u}-\widehat {\bf{u}}^L,\boldsymbol{u}-\widehat {\bf{u}}^L).$$ In the second example, we consider the case where $$b$$ is the identity matrix, i.e., it does not depend on $$y$$. In this case, from (2.10) we note that the function $${\frak u_1}=0$$. We choose   a(x,y)=(1+x1)(1+x2)(1+cos2⁡2πy1)(1+cos2⁡2πy2) and   f=(4(2π(1+x1)(1+x2)sin⁡2πx2+(1+x1)(cos⁡2πx1−cos⁡2πx2))9+12πsin⁡2πx24(2π(1+x1)(1+x2)sin⁡2πx1−(1+x2)(cos⁡2πx1−cos⁡2πx2))9+12πsin⁡2πx1) so that the solution to the homogenized problem is   u0=(12πsin⁡2πx212πsin⁡2πx1). Figure 2 plots the energy error versus the mesh size for the sparse tensor product FE approximations for the two-scale homogenized Maxwell-type problem. The plot confirms the analysis. Fig. 2. View largeDownload slide The sparse tensor energy error $$B(\boldsymbol{u}-\widehat {\bf{u}}^L,\boldsymbol{u}-\widehat {\bf{u}}^L).$$ Fig. 2. View largeDownload slide The sparse tensor energy error $$B(\boldsymbol{u}-\widehat {\bf{u}}^L,\boldsymbol{u}-\widehat {\bf{u}}^L).$$ Acknowledgements The authors gratefully acknowledge a postgraduate scholarship of Nanyang Technological University, the AcRF Tier 1 grant RG69/10, the Singapore A*Star SERC grant 122-PSF-0007 and the AcRF Tier 2 grant MOE 2013-T2-1-095 ARC 44/13. Footnotes 1 The notations $$Y_1,\ldots,Y_n$$, which denote the same unit cube $$Y$$, are introduced for convenience only, especially in the case where the Cartesian product of several of them is used, to avoid the necessity of indicating how many times the unit cube appears in the product. The functions $$a$$ and $$b$$ depend on the macroscopic scale only and are periodic with respect to $$y_i$$ with the period being the unit cube $$Y$$. 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( 2010) Multiscale computations for 3D time-dependent Maxwell’s equations in composite materials. SIAM J. Sci. Comput. , 32, 2560– 2583. Google Scholar CrossRef Search ADS   Appendix A. We present the proof of Theorem 4.2 in this appendix. We consider a set of $$M$$ open cubes $$Q_i$$ ($$i=1,\ldots,M$$) of size $$\varepsilon^t$$ for $$t>0$$ to be chosen later such that $$D\subset\bigcup_{i=1}^MQ_i$$ and $$Q_i\bigcap D\ne\emptyset$$. Each cube $$Q_i$$ intersects with only a finite number, which does not depend on $$\varepsilon$$, of other cubes. We consider a partition of unity that consists of $$M$$ functions $$\rho_i$$ such that $$\rho_i$$ has support in $$Q_i$$, $$\sum_{i=1}^M\rho_i(x)=1$$ for all $$x\in D$$ and $$|\nabla\rho_i(x)|\le c\varepsilon^{-t}$$ for all $$x$$ (indeed such a set of cubes $$Q_i$$ and a partition of unity can be constructed from a fixed set of cubes of size $${\mathcal O}(1)$$ by rescaling). For $$r=1,2,3$$ and $$i=1,\ldots,M$$, we define   Uir=1|Qi|∫Qicurlu0(x)rdx and   Vir=1|Qi|∫Qiu0(x)rdx (as $$u_0\in H^s(D)^3$$ and $${\rm curl\,} u_0\in H^s(D)^3$$, for the Lipschitz domain $$D$$, we can extend each of them, separately, continuously outside $$D$$ and understand $$u_0$$ and $${\rm curl\,} u_0$$ as these extensions; see Wloka, 1987, Theorem 5.6). Let $$U_i$$ and $$V_i$$ denote the vectors $$(U_i^1, U_i^2,U_i^3)$$ and $$(V_i^1,V_i^2,V_i^3),$$ respectively. Let $$B$$ be the unit cube in $$\mathbb{R}^3$$. From the Poincaré inequality, we have   ∫B|ϕ−∫Bϕ(x)dx|2dx≤c∫B|∇ϕ(x)|2dx  ∀ϕ∈H1(B). By translation and scaling, we deduce that   ∫Qi|ϕ−1|Qi|∫Qiϕ(x)dx|2dx≤cε2t∫Qi|∇ϕ(x)|2dx  ∀ϕ∈H1(Qi), i.e.,   ‖ϕ−1|Qi|∫Qiϕ(x)dx‖L2(Qi)≤cεt‖ϕ‖H1(Qi). Together with   ‖ϕ−1|Qi|∫Qiϕ(x)dx‖L2(Qi)≤c‖ϕ‖L2(Qi), we deduce from interpolation that   ‖ϕ−1|Qi|∫Qiϕ(x)dx‖L2(Qi)≤cεts‖ϕ‖Hs(Qi)  ∀ϕ∈Hs(Qi). Thus,   ∫Qi|curlu0(x)r−Uir|2dx≤cε2ts‖(curlu0)r‖Hs(Qi)2. (A.1) Let   u1ε(x)=u0(x)+εNr(x,xε)Ujrρj(x)+ε∇[wr(x,xε)Vjrρj(x)]. We have   curl(aε(x)curlu1ε(x))+bε(x)u1ε(x) =curla(x,xε)[curlu0(x)+εcurlxNr(x,xε)Ujrρj(x)+curlyNr(x,xε)Ujrρj+ε(Ujr∇ρj)×Nr(x,xε)] +b(x,xε)[u0(x)+εNr(x,xε)Ujrρj(x)+ε∇xwr(x,xε)Vjrρj(x)+∇ywr(x,xε)Vjrρj(x)  + εwr(x,xε)Vjr∇ρj(x)] =curl(a0(x)curlu0(x))+b0(x)u0(x)+curl[Gr(x,xε)Ujrρj(x)]+gr(x,xε)Vjrρj(x)+εcurlI(x) +εJ(x)+curl[(aε(x)−a0(x))(curlu0(x)−Ujρj(x))]+(bε(x)−b0(x))(u0(x)−Vjρj(x)), where the vector functions $$G_r(x,y)$$ and $$g_r(x,y)$$ are defined by   (Gr)i(x,y) =air(x,y)+aij(x,y)curlyNr(x,y)j−air0(x), (A.2)  (gr)i(x,y) =bir(x,y)+bij(x,y)∂wr∂yj(x,y)−bir0(x) (A.3) and   I(x) =a(x,xε)[curlxNr(x,xε)Ujrρj(x)+(Ujr∇ρj(x))×Nr(x,xε)],J(x) =b(x,xε)[Nr(x,xε)Ujrρj(x)+∇xwr(x,xε)Vjrρj(x)+wr(x,xε)Vjr∇ρj(x)]. Therefore, for $$\phi\in W$$,    ⟨curl(aεcurlu1ε)+bεu1ε−curl(a0curlu0)−b0u0,ϕ⟩  =∫DUjrρj(x)Gr(x,xε)⋅curlϕdx+∫DVjrρj(x)gr(x,xε)⋅ϕ(x)dx  +ε∫DI(x)⋅curlϕ(x)dx+ε∫DJ(x)⋅ϕ(x)dx+∫D(aε−a0)(curlu0(x)−Ujρj)⋅curlϕ(x)dx +∫D(bε−b0)(u0−Vjρj)⋅ϕdx (here $$\langle\cdot\rangle$$ denotes the duality pairing between $$W'$$ and $$W$$). From (4.4), we have that $${\rm curl}_yG_r(x,y)=0$$. Further, from (4.6) $$\int_YG_r(x,y)\,{\rm d}y=0$$. Therefore, there is a function $$\tilde G_r(x,y)$$ such that $$G_r(x,y)=\nabla_y \tilde G_r(x,y)$$. From (4.1), we have $${\rm div}_yg_r(x,y)=0$$ and from (4.6) $$\int_Yg_r(x,y)\,{\rm d}y=0$$. Hence, there is a function $$\tilde g_r$$ such that $$g_r(x,y)={\rm curl}_y\tilde g_r(x,y)$$. As $$\nabla_y\tilde G_r(x,\cdot)=G_r(x,\cdot)\in H^1(Y)^3$$ so $$\Delta_y\tilde G_r(x,\cdot)\in L^2(Y)$$. Thus, $$\tilde G_r(x,\cdot)\in H^2(Y),$$ which implies $$\tilde G_r(x,\cdot)\in C(\bar Y)$$. As $$G_r(x,\cdot)\in C^1(\bar D,H^1_\#(Y)^3)$$, we deduce that $$\tilde G_r(x,y)\in C^1(\bar D, H^2(Y))\subset C^1(\bar D,C(\bar Y))$$. The construction of $$\tilde g_r$$ in Jikov et al. (1994) implies that $$\tilde g_r\in C^1(\bar D, C(\bar Y))$$ (see Hoang & Schwab, 2013). We have    ∫DUjrρjGr(x,xε)⋅curlϕdx=∫DUjrρj(x)[ε∇G~r(x,xε)−ε∇xG~r(x,xε)]⋅curlϕdx =−ε∫DG~r(x,xε)div[(Ujrρj)curlϕ]dx−ε∫DUjrρj∇xG~r(x,xε)⋅curlϕdx. We note that   |∫DUjrρj∇xG~r(x,xε)⋅curlϕdx|≤c‖(Ujrρj)‖L2(D)‖curlϕ‖L2(D)3. From   ‖Ujrρj‖L2(D)2=∫D(Ujr)2ρj(x)2dx+∑i≠j∫DUirUjrρi(x)ρj(x)dx, and the fact that the support of each function $$\rho_i$$ intersects only with the support of a finite number (which does not depend on $$\varepsilon$$) of other functions $$\rho_j$$ in the partition of unity, we deduce   ‖Ujrρj‖L2(D)2 ≤c∑j=1M(Ujr)2|Qj| =c∑j=1M1|Qj|(∫Qjcurlu0(x)rdx)2≤c∑j=1M∫Qjcurlu0(x)r2dx≤c∫Dcurlu0(x)r2dx. Thus,   |ε∫DUjrρj∇xG~r(x,xε)⋅curlϕdx|≤cε‖curlϕ‖L2(D)3. We have further that   ε∫DG~r(x,xε)div[(Ujrρj)curlϕ]dx =ε∫DG~r(x,xε)[(Ujr∇ρj(x))]⋅curlϕdx ≤cε‖Ujr∇ρj‖L2(D)3‖curlϕ‖L2(D)3. As the support of each function $$\rho_i$$ intersects with the support of a finite number of other functions $$\rho_j$$ and $$\|\nabla\rho_j\|_{L^\infty(D)}\le c\varepsilon^{-t}$$, we have   ‖Ujr∇ρj‖L2(D)32≤c∑j=1M(Ujr)2|Qj|‖∇ρj‖L∞(D)2≤cε−2t∑j=1M(Ujr)2|Qj|≤cε−2t, so   ε∫DG~r(x,xε)div[(Ujrρj)curlϕ]dx≤cε‖Ujr∇ρj‖L2(D)3‖curlϕ‖L2(D)3≤cε1−t‖curlϕ‖L2(D)3. We therefore deduce that   |∫DUjrρjGr(x,xε)⋅curlϕdx|≤cε1−t‖curlϕ‖L2(D)3. We have   ∫DVjrρjgr(x,xε)⋅ϕ(x)dx=∫DVjrρj[εcurlg~r(x,xε)−εcurlxg~r(x,xε)]⋅ϕdx. Arguing similarly to above, we have   |ε∫DVjrρjcurlxg~r(x,xε)⋅ϕdx|≤cε‖Vjrρj‖L2(D)3‖ϕ‖L2(D)3≤cε‖ϕ‖L2(D)3 and   |ε∫DVjrρjcurlg~r(x,xε)⋅ϕdx| = |ε∫Dg~r(x,xε)⋅curl[(Vjrρj)ϕ]dx| ≤ |ε∫Dg~r(x,xε)⋅[(Vjrρj)curlϕ+ϕ×(Vjr∇ρj)]dx ≤c(ε‖curlϕ‖L2(D)3+cε1−t‖ϕ‖L2(D)3)(∑j=1M(Vjr)2|Qj|)1/2 ≤c(ε‖curlϕ‖L2(D)3+cε1−t‖ϕ‖L2(D)3). We note that   ‖I‖L2(D)3≤csupr[‖Ujrρj‖L2(D)+‖Ujr∇ρj‖L2(D)]≤cε−t and   ‖J‖L2(D)3≤csupr[‖Ujrρj‖L2(D)+c‖Vjrρj‖L2(D)+c‖Vjr∇ρj‖L2(D)]≤cε−t. We have further that   ⟨curl((aε−a0)(curlu0−Ujρj)),ϕ⟩≤c‖curlu0−(Ujρj))‖L2(D)3‖curlϕ‖L2(D)3. From   ∫D|(curlu0)r−(Ujrρj)|2dx=∫D|∑j=1M((curlu0)r−Ujr)ρj|2dx, using the support property of $$\rho_j$$, we have from (A.1),   ∫D|(curlu0)r−(Ujrρj)|2dx ≤c∑j=1M∫Qj|(curlu0)r−Ujr|2dx≤cε2st∑j=1M‖(curlu0)r‖Hs(Qj)2 =cε2st∑j=1M[∫Qj(curlu0)r2dx+∫Qj×Qj(curlu0(x)r−curlu0(x′)r)2|x−x′|3+2sdxdx′] ≤cε2st[‖(curlu0)r‖L2(D)2+∫D×D(curlu0(x)r−curlu0(x′)r)2|x−x′|3+2sdxdx′] =cε2st‖(curlu0)r‖Hs(D)2. (A.4) Thus,   ⟨curl((aε−a0)(curlu0−Ujρj)),ϕ⟩≤cεst‖curlϕ‖L2(D)3. Similarly, we have   |∫D(bε−b0)(u0−∑j=1MVjρj)⋅ϕdx|≤c‖∑j=1M(u0−Vj)ρj‖L2(D)3‖ϕ‖L2(D)3≤cεst‖ϕ‖L2(D)3. Therefore,   |⟨curl(aεcurlu1ε)+bεu1ε−curl(a0curlu0)−b0u0,ϕ⟩|≤c(ε1−t+εst)‖ϕ‖V i.e.,   ‖curl(aεcurlu1ε)+bεu1ε−curl(a0curlu0)−b0u0‖W′≤c(ε1−t+εst). Thus,   ‖curl(aεcurlu1ε)+bεu1ε−curl(aεcurluε)−bεuε‖W′≤c(ε1−t+εst). (A.5) Let $$\tau^\varepsilon(x)$$ be a function in $$\mathcal{D}(D)$$ such that $$\tau^\varepsilon(x)=1$$ outside an $$\varepsilon$$ neighbourhood of $$\partial D$$ and $${\rm sup}_{x\in D}\varepsilon|\nabla\tau^\varepsilon(x)|<c,$$ where $$c$$ is independent of $$\varepsilon$$. We consider the function   w1ε(x)=u0(x)+ετε(x)Ujrρj(x)Nr(x,xε)+ε∇[Vjrρjτε(x)wr(x,xε)]. We then have   u1ε−w1ε=ε(1−τε(x))Ujrρj(x)Nr(x,xε)+ε∇[(1−τε(x))Vjrρjwr(x,xε)] and   curl(u1ε−w1ε) =εcurlxNr(x,xε)Ujrρj(x)(1−τε(x))+curlyNr(x,xε)Ujrρj(x)(1−τε(x)) −εUjrρj(x)∇τε(x)×Nr(x,xε)+ε(1−τε(x))Ujr∇ρj(x)×Nr(x,xε). As shown above, $$\|U_j^r\rho_j\|_{L^2(D)}$$ is uniformly bounded, so   ‖εcurlxNr(x,xε)(Ujrρj)(1−τε(x))‖L2(D)3≤cε. Let $$\tilde D^\varepsilon$$ be the $$3\varepsilon^{t}$$ neighbourhood of $$\partial D$$. We note that $${\rm curl\,} u_0$$ is extended continuously into a function in $$H^s(\mathbb{R}^3)$$ outside $$D$$. As shown in Hoang & Schwab (2013), for $$\phi\in H^1(\tilde D^\varepsilon)$$,   ‖ϕ‖L2(D~ε)≤cεt/2‖ϕ‖H1(D~ε). From this and   ‖ϕ‖L2(D~ε)≤‖ϕ‖L2(D~ε), using interpolation, we get   ‖ϕ‖L2(D~ε)≤cεst/2‖ϕ‖Hs(D~ε)≤cεst/2‖ϕ‖Hs(D) for all $$\phi\in H^s(D)$$ extended continuously outside $$D$$. We then have   ‖Ujrρj‖L2(Dε)2 ≤c∑j=1M∫Qj⋂Dε(Ujr)2ρj2dx ≤c∑j=1M|Qj⋂Dε|1|Qj|2(∫Qj(curlu0)rdx)2 ≤c∑Qj⋂Dε≠∅|Qj⋂Dε||Qj|∫Qj(curlu0)r2dx. As $$D^\varepsilon$$ is the $$\varepsilon$$ neighbourhood of $$\partial D$$, $$\partial D$$ is Lipschitz and $$Q_j$$ has size $$\varepsilon^t$$, $$|Q_j\bigcap D^\varepsilon|\le c\varepsilon^{1+(d-1)t}$$ so $$|Q_j\bigcap D^\varepsilon|/|Q_j|\le c\varepsilon^{1-t}$$. When $$Q_j\bigcap D^\varepsilon\ne\emptyset$$, $$Q_j\subset\tilde D^\varepsilon$$. Thus,   ‖Ujrρj‖L2(Dε)2≤cε1−t‖(curlu0)r‖L2(D~ε)2≤cε1−t+st‖curlu0‖Hs(D)32. Therefore,   ‖curlyNr(x,xε)(Ujrρj)(1−τε(x))‖L2(D)3≤cε(1−t+st)/2 and   ‖ε(Ujrρj)∇τε(x)×Nr(x,xε)‖L2(D)3≤cε(1−t+st)/2. Similarly, we have   ‖Ujr∇ρj‖L2(Dε)32 ≤cε−2t∑Qj⋂Dε≠∅|Qj⋂Dε||Qj|∫Qj(curlu0)r2dx ≤cε−2t+1−t‖curlu0‖L2(D~ε)32≤cε1−3t+st‖curlu0‖Hs(D)32. Thus,   ‖ε(1−τε(x))(Ujr∇ρj)×Nr(x,xε)‖L2(D)≤cε(1−t)+(1−t+st)/2. Therefore,   ‖curl(u1ε−w1ε)‖L2(D)3≤c(ε(1−t+st)/2+ε(1−t)+(1−t+st)/2). We further have   ε∇[(1−τε(x))wr(x,xε)(Vjrρj)] = −ε∇τε(x)wr(x,xε)(Vjrρj)+ε(1−τε(x))∇xwr(x,xε)(Vjrρj) +(1−τε(x))∇ywr(x,xε)(Vjrρj)+ε(1−τε(x))wr(x,xε)(Vjr∇ρj). Arguing as above, we deduce that   ‖Vjrρj‖L2(Dε)≤cε(1−t+st)/2,  ‖Vjr∇ρj‖L2(Dε)≤cε(1−t+st)/2−t. Therefore,   ‖ε∇[(1−τε(x))wr(x,xε)(Vjrρj)]‖L2(D)3≤c(ε(1−t+st)/2+ε1−t+(1−t+st)/2). Thus,   ‖u1ε−w1ε‖L2(D)3≤c(ε(1−t+st)/2+ε(1−t)+(1−t+st)/2). Choosing $$t=1/(s+1)$$ we have   ‖curl(aεcurl(u1ε−w1ε))+bε(u1ε−w1ε)‖W′≤cεs/(s+1). This together with (A.5) gives   ‖curl(aεcurl(uε−w1ε))+bε(uε−w1ε)‖W′≤cεs/(s+1). Thus,   ‖uε−w1ε‖W≤cεs/(s+1), which implies   ‖uε−u1ε‖W≤cεs/(s+1). (A.6) We note that   curlu1ε=curlu0(x)+curlyNr(x,xε)(Ujrρj)+εcurlxNr(x,xε)(Ujrρj)+ε(Ujr∇ρj)×Nr(x,xε). From   ‖εcurlxNr(x,xε)(Ujrρj)‖L2(D)3≤cε  and  ‖ε(Ujr∇ρj)×Nr(x,xε)‖L2(D)3≤cεε−t=cεs/(1+s), we deduce that   ‖curlu1ε−curlu0−curlyNr(x,xε)(Ujrρj)‖L2(D)3≤cεs/(s+1). From (A.4),   ‖curlu0−(Ujrρj)‖L2(D)3≤cεts=cεs/(s+1), we get   ‖curlu1ε−[curlu0+curlyNr(x,xε)(curlu0)r‖L2(D)3≤cεs/(s+1). This together with (A.6) implies   ‖curluε−[curlu0+curlyNr(x,xε)(curlu0)r‖L2(D)3≤cεs/(s+1). □ Appendix B. We prove Lemma 4.3 in this appendix. We adapt the proof of Hoang & Schwab (2013, Lemma 5.5). As   u1(x,y)=∑r=13curlu0(x)rNr(x,y), it is sufficient to show that for each $$r=1,2,3$$,   ∫D|curlu0(x)rcurlyNr(x,xε)−∫Ycurlu0(ε[xε]+εt)rcurlyNr(ε[xε]+εt,xε)dt|2dx≤cε2s. The expression on the left-hand side is bounded by    ∫D∫Y|curlu0(x)rcurlyNr(x,xε)−curlu0(ε[xε]+εt)rcurlyNr(ε[xε]+εt,xε)|2dtdx ≤2∫D∫Y|(curlu0(x)r−curlu0(ε[xε]+εt)r)curlyNr(ε[xε]+εt,xε)|2dtdx +2∫D∫Y|curlu0(x)r|2|curlyNr(x,xε)−curlyNr(ε[xε]+εt,xε)|2dtdx. As $${\rm curl}_yN^r\in C^1(\bar D,C(\bar Y))^3$$, there exists a constant $$c$$ such that   supx∈Dsupt∈Y|curlyNr(x,xε)−curlyNr(ε[xε]+εt,xε)|≤cε. From this we have   ∫D|curlu0(x)rcurlyNr(x,xε)−Uε(curlu0(⋅)rcurlyNr(⋅,⋅))(x)|2dx≤c∫D∫Y|curlu0(x)r−curlu0(ε[xε]+εt)r|2dtdx+cε2. We now show that for $${\rm curl\,} u_0\in H^s(D)$$,   ∫D∫Y|curlu0(x)r−curlu0(ε[xε]+εt)r|2dtdx≤cε2s. (B.1) Letting $$\phi(x)$$ be a smooth function, we have    ∫D∫Y|ϕ(x)−ϕ(ε[xε]+εt)|2dtdx ≤∑i=1d∫D∫Y|ϕ(ε[x1ε]+εt1,…,ε[xi−1ε]+εti−1,xi,…,xd) −ϕ(ε[x1ε]+εt1,…,ε[xiε]+εti,xi+1,…,xd)|2dtdx ≤∑i=1d∫D∫Y|ε∫ti{xi/ε}∂ϕ∂xi(ε[x1ε]+εt1,…,ε[xiε]+εζi,xi+1,…,xd)dζi|2dtdx ≤ε2∑i=1d∫D∫Y∫01|∂ϕ∂xi(ε[x1ε]+εt1,…,ε[xiε]+εζi,xi+1,…,xd)|2dζidtdx ≤ε2∑i=1d∫D|∂ϕ∂xi|2dx, which follows from (4.13); here we freeze the variables $$x_{i+1},\ldots,x_d$$. Let $$\psi\in H^1(D)$$. We consider a sequence $$\{\phi_n\}_n\subset C^\infty(\bar D),$$ which converges to $$\psi$$ in $$H^1(D)$$. As $$n\rightarrow \infty$$,   ∫D∫Y(ϕn(ε[xε]+εt)−ψ(ε[xε]+εt))2dtdx = ∫DUε((ϕn−ψ)2)(x)dx ≤ ∫D(ϕn(x)−ψ(x))2dx→0. Therefore,   ∫D∫Y(ψ(x)−ψ(ε[xε]+εt))2dtdx ≤3∫D(ψ−ϕn)2dx+3∫D∫Y(ϕn−ϕn(ε[xε]+εt))2dtdx +3∫D∫Y(ϕn(ε[xε]+εt)−ψ(ε[xε]+εt))2dtdx ≤6∫D(ψ−ϕn)2dx+3ε2∑i=1d∫D|∂ϕn∂xi|2dx. Letting $$n\to\infty$$, we have   ∫D∫Y(ψ(x)−ψ(ε[xε]+εt))2dtdx≤3ε2∑i=1d∫D|∂ψ∂xi|2dx. Let $$T$$ be the linear map from $$L^2(D)$$ to $$L^2(D\times Y)$$ so that   T(ϕ)(x,y)=ϕ(x)−ϕ(ε[xε]+εt). We thus have   ‖T‖H1(D)→L2(D×Y)≤cε. On the other hand,   ‖T‖L2(D)→L2(D×Y)≤c. From interpolation theory, we deduce that   ‖T‖Hs(D)→L2(D×Y)≤cεs. We then get (B.1). The conclusion follows. □ Notes added after the proof stage: After the article is accepted, we learnt about the related recent article: P. Henning, M. Ohlberger and B. Verfürth (2016), A new heterogeneous multiscale method for time-harmonic Maxwell’s equations, SIAM J. Numer. Anal., 54, 3493–3522. This article considers a locally periodic two-scale time harmonic Maxwell equation, but the variational form is still assumed to be strictly coercive, uniformly with respect to the microscopic scale, similar to the equation considered in our present article. These authors formulate the two-scale homogenized equation in a slightly different manner. The Heterogeneous Multiscale Method (HMM) is used to solve the two-scale problem, and is shown to be equivalent to solving the two-scale homogenized equation by using the full tensor finite element spaces. © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

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IMA Journal of Numerical AnalysisOxford University Press

Published: Jan 1, 2018

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